Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
If $abc=1$ then $\sum\limits_{cyc}^{}{\frac{1}{b(a+b)}}\ge \frac{3}{2}$
If $abc=1$ for positive $a,b,c$, then $\sum\limits_{cyc}^{}{\dfrac{1}{b(a+b)}}\ge \dfrac{3}{2}$
I have tried the following,in decreasing order of success:
1)AM-GM:$a+b+c\ge 3$ and $ab+bc+ca\ge 3$
2)Substituting $1=abc$ yields nothing
3)Substituti... | Look at $\frac{1}{b(a+b)}$ and $\frac{1}{a(a+b)}$. Adding together yields $\frac{1}{ab}$. So the sum is just
$$\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ac} = \frac{a + b + c}{abc}$$
The denominator is 1 and the numerator is at least 3 by AM-GM for positive $a,b,c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1141774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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Finding number of integer solutions using Generating Functions This is a problem for a practice test my professor gave me.
$$\text{How many integer solutions are there to } x_1+x_2+x_3+x_4 \leq 50 \\ \text{with } x_i \geq 2 \text{ for all } i = 1,2,3,4 \text{ and } x_1,x_2 \leq 7 \text{?}$$
This is how I approached th... | Inclusion exclusion is easier than generating functions in this case. First you reduce to
$$ x_1 + x_2 + x_3 + x_4 + x_5 = 42 $$
where $x_i \geq 0$ and $x_1, x_2 \leq 5$. Then IE immediately gives
$$ {42+5-1 \choose 5-1} - 2{36+5-1 \choose 5-1} + {30+5-1 \choose 5-1} \\
= {46 \choose 4} - 2{40 \choose 4} + {34 \choos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Existence of two primes satisfying the given conditions I want to know whether
the equation
$x^a-x=y^b-y$
has a solution or not satisfying the conditions that $x$ and $y$ are distinct odd primes, $a$ and $b$ are integers both greater than $1$.
| Michael Bennett, On some exponential equations of S. S. Pillai, Canadian Journal of Math. 53 (2001) 897-922, reports eight solutions of $x^a-x=y^b-y$ in positive integers, with $a,b>1$:
$6=2^3-2=3^2-3$,
$30=2^5-2=6^2-6$,
$210=6^3-6=15^2-15$,
$240=3^5-3=16^2-16$,
$2184=3^7-3=13^3-13$,
$8190=2^{13}-2=91^2-91$,
$78120=5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1143521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Sum of $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots$ My problem is to find the sum of the series
$$
S = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots
$$
where the terms are the reciprocals of the positive integers whose on... | $$S = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots$$
$$\frac{1}{2}S = \frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{12}+\frac{1}{16}+\frac{1}{18}+\frac{1}{24}+\cdots$$
What terms are missing? We've lost all the terms of the form $\frac{1}{3^n}$, but have retai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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How to prove this floor function equation? How can I prove the following equation?
$$
\lfloor nx \rfloor =
\lfloor x \rfloor +
\Big\lfloor x + \frac{1}{n} \Big\rfloor +
\Big\lfloor x + \frac{2}{n} \Big\rfloor +
\Big\lfloor x + \frac{3}{n} \Big\rfloor +
\Big\lfloor x + \frac{4}{n} \Big\rfloor +
\Big\l... | If you take a closer look, you will notice that the second term within each floored term will be less than one.
E.g in $$\left\lfloor x+\frac{4}{n}\right\rfloor$$As you can see $\frac{4}{n}$ is less than one.
So, we can conclude that each term will be reduced to $\lfloor x\rfloor$ if $n$ and $x$ are integers.
We will g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Why does $\int\frac{1}{2x+1}dx=\frac{1}{2}\ln|2x+1|+C$? The way I am thinking is as follows:
$$\int\frac{1}{2x+1}\,dx = \int\frac{1}{2}\frac{1}{x+\frac{1}{2}}\,dx = \frac{1}{2}\int\frac{1}{x+\frac{1}{2}}\,dx = \frac{1}{2}\ln\left|x+\frac{1}{2}\right|+C$$
However, the textbook answer is $\frac{1}{2}\ln|2x+1|+C$. Where i... | You forgot to add a constant, $C$. This is important: your answer differs from the textbook's only by a constant.
$$\frac{1}{2}\ln\left|x + \frac{1}{2}\right| = \frac{1}{2}\ln\left|\frac{2x + 1}{2}\right| = \frac{1}{2}[\ln|2x + 1| - \ln(2)] = \frac{1}{2}\ln|2x + 1| - \frac{1}{2}\ln(2)$$
So
$$\int \frac{1}{2x+1}\, dx ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1148261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the value of the infinite sum $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + ... $ Can anyone help me to find what is the value of $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + ... $ when it tends to infinity
The first i ... | the series equal to
$$\sum_{n=0}^{\infty}\frac{1}{6n+1}-\frac{1}{6n+4}$$
the first term
$$\frac{1}{1-x^6}=\sum_{n=0}^{\infty} x^{6n}$$
$$\int_0^1\frac{1}{1-x^6}dx=\int_0^1\sum_{n=0}^{\infty}x^{6n}dx$$
$$\int_0^1\frac{1}{1-x^6}dx=\sum_{n=0}^{\infty}\frac{1}{6n+1}$$
the second term
$$\frac{1}{1-x^6}=\sum_{n=0}^{\infty}x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1148862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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The Finishing Step to Showing that $1/(1-4x)^{1/2}$ generates the sequence $\binom{2r}{r}$ The Full Question:
Show that $(1-4x)^{-\frac{1}{2}}$ generates the sequence $\binom{2n}{n}$, $n\in
\mathbb N$
My Research
How to show that $1 \over \sqrt{1 - 4x} $ generate $\sum_{n=0}^\infty \binom{2n}{n}x^n $
Show $\sum\limits... |
Note: Avoiding a small calculation error in OPs work in 4.) and considering a small trick in 6.) will close the gap. Here's a calculation:
\begin{align*}
\binom{-\frac{1}{2}}{r}
&=\frac{\left(-\frac{1}{2}\right)\left(-\frac{1}{2}-1\right)\left(-\frac{1}{2}-2\right)\cdot\ldots\cdot
\left(-\frac{1}{2}-(r-1)\right)}{r!}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1151233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root. If it is known that the equation
$$ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $$
has a (real) root, prove the inequality
$$ a^2 + b^2 \geq 8. $$
I am stuck on this problem, though, it is a very easy problem for my math teacher. Anyway,... | I assume that $a,b$ ar real.
Suppose that $a^2+b^2<8$ and the polynomial has the real root $\xi$. It follows that:
$$
0=\xi^4 + a\xi^3 + 2\xi^2 + b\xi + 1>\xi^4 + a\xi^3 + \frac{a^2+b^2}{4}\xi^2 + b\xi + 1=\xi^2\left(\xi+\frac{a}{2}\right)^2+\left(\frac{b}{2}\xi+1\right)^2
$$
But the sum of squares of real numbers is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1151480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Deriving an expression for $\cos^4 x + \sin^4 x$
Derive the identity $\cos^4 x + \sin^4 x=\frac{1}{4} \cos (4x) +\frac{3}{4}$
I know $e^{i4x}=\cos (4x) + i \sin (4x)=(\cos x +i \sin x)^4$. Then I use the binomial theorem to expand this fourth power, and comparing real and imaginary parts, I conclude that $\cos^4 x + ... | $$\begin{align}
\cos^4x + \sin^4 x &= \left( \cos^2 x \right)^2 + \left( \sin^2 x \right)^2 \\[4pt]
&= \left( \frac{1 + \cos 2 x}{2}\right)^2 + \left( \frac{1-\cos 2x}{2} \right)^2 \\[4pt]
&= \frac{1}{2}\left( 1 + \cos^2 2 x \right) \\[4pt]
&= \frac{1}{2}\left( 1 + \frac{1+\cos 4 x}{2} \right) \\[4pt]
&= \frac{1}{4}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1153131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
Is there a simple expression for ${}_2F_1(a,a+\tfrac{1}{2};a+1;z)$? I have been searching through some books and also this
but I have not succeed.
I wonder if there is a simple equivalent form for ${}_2F_1(a,a+\tfrac{1}{2};a+1;z)$, in terms of elementary functions or other simpler functions.
In principle, $a>0$ but a s... | For the sake of simplicity, I'll assume $z$ is real and $0 < a \le 1/2$. First, suppose $a = 1/2$ and $z\neq 0,1$. Then
\begin{align}{}_2F_1\left(a, a+\frac{1}{2}; a + 1; z\right) &= {}_2F_1\left(\frac{1}{2},1;\frac{3}{2};z \right)\\
&= \frac{1}{i\sqrt{z}}\cdot i\sqrt{z}\,{}_2F_1\left(1,\frac{1}{2};\frac{3}{2};-(i\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Implicit differentiation of $(x^2 + y^2)^2 = (x-y)^2$ $$
(x^2 + y^2)^2 = (x-y)^2$$
Wolfram alpha yields this answer:
$$
y'(x) = \frac{(-2 x^3-2 x y^2+x-y)}{((2 x^2-1) y+x+2 y^3)}$$
But it's impossible to get $-y$ in the denominator
Actually, my answer is pretty much the same, except it's $+ y$ in the denominator
When d... | You start off with
$$
(x^2+y^2)^2=(x-y)^2 \Longleftrightarrow x^4+2x^2y^2+y^4=x^2-2xy+y^2.
$$
Now pull everything with a $y$ in it to the LHS and terms only with $x$ to the RHS:
$$
y^4-y^2+2x^2y^2+2xy=x^2-x^4.
$$
Now implicitly differentiation everything (using the product rule twice):
$$
4y^3y'-2yy'+(4xy^2+4x^2yy')+(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1161323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to factorise $x^4 - 3x^3 + 2$, so as to compute the limit of a quotient? Question:
Find the limit: $$\lim_{x \to 1}\frac{x^4 - 3x^3 + 2}{x^3 -5x^2+3x+1}$$
The denominator can be simplified to: $$(x-1)(x^2+x)$$
However, I am unable to factor the numerator in a proper manner (so that $(x-1)$ will cancel out)
I know... | You may use Horner here. Since $x=1$ is a root of the nominator (you can check that easily) then you can deduce that down by applying a long division.
Hence your limit is deduced down to:
$$\lim_{x\rightarrow 1}\frac{x^4-3x^3+2}{x^3-5x^2+3x+1}=\lim_{x\rightarrow 1}\frac{(x-1)\left ( x^3-2x^2-2x-2 \right )}{\left ( x-1 ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $a\leq (b + c)/2$ with $a,b,c>0$, why $a^2\leq (b^2 + c^2)/2$? If $a\leq (b + c)/2$ with $a,b,c>0$, why $a^2\leq \frac{b^2 + c^2}{2}$? I can only see how to get $a^2\leq \frac{b^2+c^2 + 2ab}{4}$.
| because $(a+b)^2 \neq a^2+b^2.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1165071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$. Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.
My work: So I think I have to do a proof by induction and I just wanted some help editing my proof.
My attempt:
Let $P(n)=1^2+2^2+\cdots+n^2=\frac{n(n+1)... | Your inductive assumption is such that the formula marked $\color{red}{\mathrm{red}}$ (several lines below) holds for $i=k$:
$$\sum^{i=k}_{i=1} i^2=\frac{k(k+1)(2k+1)}{6} $$
You need to prove that for $i=k+1$: $$\sum^{i=k+1}_{i=1} i^2=\color{blue}{\frac{(k+1)(k+2)(2k+3)}{6}}$$
To do this you cannot use: $$\sum^{i=k}_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 2
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Modulo polynomial in ring theory Let $x^4-16$ be an element of the polynomial ring $E= \mathbb{Z}[x]$ and use the bar notation to denote passage to the quotient ring $\mathbb{Z}[x]/(x^4-16)$. Find a polynomial of degree $\leq 3$ that is congruent to $7x^{13} -11x^9 + 5x^5-2x^3+3$ modulo $x^4-16$. Can anyone help unders... | ${\rm mod}\,\ x^4-2^4\!:\,\ x^4\equiv 2^4\,\Rightarrow\, \color{#c00}{x^{4n}}\equiv 2^{4n}$
$\! \begin{align}{\rm therefore}\quad &\ \ 7\,\color{#c00}{x^{12}}x -11\,\color{#c00}{x^8}\,x + 5\,\color{#c00}{x^4}\,x-2x^3+3\\
\equiv &\ \ 7\, 2^{12}\,x - 11\, 2^8\, x + 5\, 2^4\, x -2\,x^3+3\quad {\bf QED}\end{align}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1166851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $PF.PG=b^2$ in a hyperbola If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G' and CF be the perpendicular to the normal from the center C then show that $$PF.PG=b^2\space and \space PF.PG'=a^2.$$ We know that the equation of th... | $\rm PF.PG$ = Power of P wrt the circle with CG as diameter, the equation of whose is:
$$x(x-(1/a)(a^2+b^2)\sec\theta)+y^2=0$$
So, $${\rm PF.PG}=|a\sec\theta(a\sec\theta-(1/a)(a^2+b^2)\sec\theta)+b^2\tan^2\theta|=b^2$$
Similiarly:
$\rm PF.PG'$ = Power of P wrt the circle with CG' as diameter, the equation of whose is:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1167958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Use $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$ to compute $\sum_{n=1}^\infty \frac{(-1)^n}{n^4}$ Is it possible to use the fact that $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$ to compute $\sum_{n=1}^\infty \frac{(-1)^n}{n^4}$?
| Hint:
Let
$$E=\sum_{n=1, n=even}^\infty \frac{1}{n^4}$$
$$O=\sum_{n=1, n=odd}^\infty \frac{1}{n^4}$$
You are given that $E+O=\frac{\pi^4}{90}$ and want to find $E-O$.
To do this note that
$$E=\sum_{n=1, n=even}^\infty \frac{1}{n^4}=\sum_{k=1}^\infty \frac{1}{(2k)^4}=\frac{1}{16}\sum_{k=1}^\infty \frac{1}{k^4}=\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1169861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Finding square root of $-5-12i$ by formula and by De Moivre's Theorem I was trying to obtain the square root of $-5-12i$ by the formula for square root (given below) and also by De Moivre's theorem and verify that both give the same result. But the two results are somehow not matching for this complex number. I am writ... | Toby's answer has a minor miscue. Also, in the first chapter of Bruce Palka's "An Introduction To Complex Function Theory", an easier algorithm is given.
Let $\;z = -5 - 12i,\;$ and let $\alpha = \text{Arg}(z).$ This means that $\;\alpha \,\in (-\pi,\pi]\;$ and $\;z = |z|[\cos(\alpha) + i\sin(\alpha)].$
Further, let ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1171492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $x^3-y^3 = (x-y)(x^2+xy+y^2)$ without expand the right side? I can prove that $x^3-y^3 = (x-y)(x^2+xy+y^2)$ by expanding the right side.
*
*$x^3-y^3 = (x-y)x^2 + (x-y)(xy) + (x-y)y^2$
*$\implies x^3 - x^2y + x^2y -xy^2 + xy^2 - y^3$
*$\implies x^3 - y^3$
I was wondering what are other ways to prove... | $x^3-y^3=x^2(x-y)+yx^2-y^3=x^2(x-y)+yx(x-y)+xy^2-y^3=...$ (Simly insert $x^2y-yx^2$, then insert $y^2x-xy^2$,...)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1172119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 12,
"answer_id": 7
} |
Indefinite integral question: $\int \frac{1}{x\sqrt{x^2+x}}dx$ How can I solve this integral: $$\int \frac{1}{x\sqrt{x^2+x}}dx$$ I first completed the square and got: $$\int \frac{1}{x\sqrt{(x+\frac{1}{2})^2-\frac{1}{4}}}dx$$ Then I factored out 1/4 and got: $$2\int \frac{1}{x\sqrt{(2x+1)^2-1}}dx$$ Then I substituted $... | Here is an approach.
$$
\begin{align}
\int \frac{1}{x \sqrt{x^2+x}}dx &=\int \frac{1}{\sqrt{1+\frac{1}{x}}}\frac{dx}{x^2}\\\\
&=-\int \frac{1}{\sqrt{1+u}}\:du \quad (u=1/x)\\\\
&=-2\sqrt{u+1}+C\\\\
&=-2\sqrt{\frac{x+1}{x}}+C
\end{align}
$$ where $C$ is any constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1172599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
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Critical numbers of the function: $x\sqrt{5-x}$ Let f(x) = $$\displaystyle f(x) = x\sqrt{5-x} $$
On the interval: [-6,4]
Critical numbers are the the values of x in the domain of f for which f'(x) = 0 or f'(x) is undefined.
Derivative of the function:
$$ \frac{1}{2} \cdot x (5-x)^{\frac{-1}{2}} \cdot -1$$
$$ \frac {\fr... | it would be much easier to work with $$y^2 = x^2(5-x), x \le 5.$$ taking the difference we find $$2y \, dy = (10x - 3x^2)\, dx $$ setting $\frac{dy}{dx} = 0$ gives you $$x = 0,\, x = \frac{10}3.$$ the spurious critical number $x = 0$ is an artifact of squaring. we know that $y = x \sqrt 5+\cdots$ for $x = 0+\cdots.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1175845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Multiplication and Addition tables the following: What would be the addition and multiplication tables of $Z_2[x]/\langle x^2 + x\rangle$?
I know how to do the addition and multiplication tables for normal modular arithmetic, butam not sure about this.
| The tables are
$$
\begin{array}{r|c c c c}
+ & 0 & 1 & x & 1+x \\
\hline
0 & 0 & 1 & x & 1+x \\
1 & 1 & 0 & 1+x & x \\
x & x & 1+x & 0 & 1 \\
1 + x & 1+x & x & 1 & 0
\end{array}
$$
and
$$
\begin{array}{r|c c c c}
\times & 0 & 1 & x & 1+x \\
\hline
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & x & 1+x \\
x & 0 & x & x & 1 \\
1 + x... | {
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"timestamp": "2023-03-29T00:00:00",
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} |
Prove by induction that $3^{3n+1} + 2^{n+1}$ is divisible by 5 How do I do this? I've tried using logarithms, factoring, but nothing seems to work.
| Other users have already outlined the proof by induction, but I think a direct proof is interesting as well.
By Fermat's little theorem (or by inspection), we know that
$$3^4 \equiv 2^4 \equiv 1 \pmod{5}$$
This means that powers of $3$, modulo $5$, are determined by the exponents remainder when divided by $4$.
$$3^{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1176215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 4
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If ${a}$ is an arbitrary integer, then prove that ${360|a^2(a^2-1)(a^2-4)}$. I think I have solved the problem. I want to verify my proof, since I don't have a teacher to help me.
Proof:
Since, ${360=8*45}$ and ${gcd(45,8)=1}$, hence if we can prove that ${45|a^2(a^2-1)(a^2-4)}$ and ${8|a^2(a^2-1)(a^2-4)}$ , then we a... | Perhaps not as formal, but easy to follow if you look at it like this:
$$360 = 2^3 \cdot 3^2 \cdot 5$$
and
$$a^2(a^2-1)(a^2-4) = (a-2)(a-1)a^2(a+1)(a+2)$$
Notice that this is a product of five consecutive integers (with the middle one squared), so:
*
*certainly one of the factors will be divisible by 5
*either $a$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1176779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Given the $ x+y+z =3$ Prove that $ x^2+y^2+z^2 \geq 3xyz$ Given the $ x+y+z =3$ and x, y and z are positive numbers. How to prove that $ x^2+y^2+z^2 \geq 3xyz$.
I tried many methods but I failed.
I did the AM-HM in-equality, but failed.
$\frac{\frac{x}{yz} + \frac{y}{zx} + \frac{z}{xy}}{3} \geq \frac{3}{\frac{xy}{z}+... | LHS= $(x^2+y^2+z^2)(\frac {x+y+z} {3}) \geq 3(xyz)^{\frac {2} {3}}.(xyz)^{\frac {1} {3}}$=RHS. (I have used here AM-GM inequality twice.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Let $\theta=\frac{2 \pi}{67}$ consider the rotation matrix $A$. What is $A^{2010}$?
Let $\theta=\frac{2 \pi}{67}$. Consider the matrix
$$A = \begin{pmatrix}
\cos\theta & \sin\theta\\
-\sin \theta& \cos \theta
\end{pmatrix} $$
Then the matrix $A^{2010}$ is?
My approach
$$A^2 = \begin{pmatrix}
\cos2\theta & \s... | Observe that when you multiply the matrices
$$\begin{pmatrix}
\cos\theta & \sin\theta\\
-\sin \theta& \cos \theta
\end{pmatrix} *
\begin{pmatrix}
\cos\phi & \sin\phi\\
-\sin \phi& \cos \phi
\end{pmatrix}
=
\begin{pmatrix}
\cos(\theta+\phi) & \sin(\theta+\phi)\\
-\sin (\theta+\phi)& \cos (\theta+\phi)
\end{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Integrals, arc length I found this on my test today, and i didn't manage to solve it.
My question is.
How to find arc length of $f(x) = x^3/3$ from $x=1$ to $x=2$?
If I use formula for arc length which is $\ell = \int_a^b \sqrt{1 + (f'(x))^2} \, dx$ where is $f'(x)=x^2$ I will get $\ell = \int_1^2 \sqrt{1+x^4} \, dx$
I... | We have:
$$\ell = \int_{1}^{2}x\,\sqrt{x^2+\frac{1}{x^2}}\,dx=\int_{1}^{2}x\,\sqrt{\left(x-\frac{1}{x}\right)^2+2}\,dx$$
and by replacing $x-\frac{1}{x}$ with $t$ we have:
$$\ell = \frac{1}{4}\int_{0}^{3/2}\left(t+\sqrt{4+t^2}\right)^2\sqrt{\frac{2+t^2}{4+t^2}}\,dt$$
hence:
$$\ell = \frac{17\sqrt{17}}{48}-\frac{\sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1179639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What are the ways to solve this integral? $$\int _{0}^{\frac{\pi}{2}}\frac{1}{5\sin x+3} dx.$$
I've tried the way of let $u=\tan \left( \frac{x}{2}\right) $ , but it's very complicated.
| No it's not very complicated, proceed carefully:
$$\scriptsize I=\int\frac{{\rm d}x}{3+5\sin x}=\int\frac{{\rm d}x}{3+5\left(\frac{2\tan(x/2)}{1+\tan^2(x/2)}\right)}=\int\frac{(1+\tan^2(x/2)){\rm d}x}{3(1+\tan^2(x/2))+10\tan (x/2)}=\int\frac{\sec^2(x/2){\rm d}x}{3\tan^2(x/2)+10\sin x+3}$$
Now, take $y=\tan(x/2)\implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1183550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Start from $259^2 + 1^2 = 34 \cdot 1973$ and use the descent procedure to write the prime 1973 as a sum of two squares.
Start from $259^2 + 1^2 = 34 \cdot 1973$ and use the descent procedure to write the prime 1973 as a sum of two squares.
How to solve it using fermat descent method?
| $34 \cdot 1973 = 259^2 + 1^2; \\ (21^2 + 1^2)(259^2 + 1^2)=(21 \cdot 259 +1)^2+(259-21)^2 \implies 13 \cdot 1973 = 7^2 + 160^2. $
$(7^2 + 4^2)(7^2 + 160^2)=(7 \cdot 7 +160 \cdot 4)^2+(160 \cdot 7-4 \cdot 7)^2 \implies 5 \cdot 1973 = 53^2 + 84^2. $
$(2^2 + 1^2)(53^2 + 84^2)=(2 \cdot 53 + 84 \cdot 1 )^2+(84 \cdot 2 -53... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1183726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$ I have found myself faced with evaluating the following integral: $$\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx. $$
Mathematica gives a closed form of $8 \pi ^3/(81 \sqrt{3})$, but I have no idea how to arrive at this close... | Shocked, shocked! that there is no contour integration yet. So, without further ado...
Note that
$$f(x) = \frac{\log^2{x}}{x^2+x+1} \implies f \left ( \frac1{x} \right ) = x^2 f(x) $$
Thus,
$$\int_1^{\infty} dx \frac{\log^2{x}}{x^2+x+1} = \int_0^{1} \frac{\log^2{x}}{x^2+x+1} = \frac12 \int_0^{\infty} dx \frac{\log^2{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1186002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Evaluation of $\int\frac{1}{x^4-5x^2+16}dx$
Evaluation of $\displaystyle \int\frac{1}{x^4-5x^2+16}\,dx$
$\bf{My\; Try::}$ Given $$\displaystyle \int\frac{1}{x^4-5x^2+16}dx = \frac{1}{8}\int\frac{\left(x^2+4\right)-\left(x^2-4\right)}{x^4-5x^2+16}\,dx$$
So We get $$\displaystyle = \frac{1}{8}\int\frac{x^2+4}{x^4-5x^2+... | Factor the denominator and expand the rational function into partial fractions:
\begin{eqnarray*}
\frac{1}{x^{4}-5x^{2}+16} &=&\frac{1}{\left( x^{2}-\sqrt{13}x+4\right)
\left( x^{2}+\sqrt{13}x+4\right) } \\
&=&\frac{1}{104}\frac{\sqrt{13}x+13}{x^{2}+\sqrt{13}x+4}-\frac{1}{104}\frac{
\sqrt{13}x-13}{x^{2}-\sqrt{13}x+4}.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1186356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Center of mass of Right angle Trapezoid Given bases $a$, $b$, and the height $h$.
Get the $M(x,y)$ coordinates formula from point $O(0,0)$, where $M$ is center of mass. Wiki has a formula for $M(y) = \frac{h}{3}\frac{2a+b}{a+b}$. And I'm interested in how to find that formula (also for $M(x)$).
The median $c$ is divid... | 1190922
$x=\frac{{ \begin{vmatrix}0&0&1\\a&0&1\\0&h&1\\\end{vmatrix} · (0+a+0) }+{ \begin{vmatrix}b&h&1\\0&h&1\\a&0&1\\\end{vmatrix} · (b+0+a) }}{ 3\left(\begin{vmatrix}0&0&1\\a&0&1\\0&h&1\\\end{vmatrix} + \begin{vmatrix}b&h&1\\0&h&1\\a&0&1\\\end{vmatrix}\right) }$
$y=\frac{{ \begin{vmatrix}0&0&1\\a&0&1\\0&h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1190922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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eigenvectors of a matrix with known eigenvalues $$
\begin{matrix}
1 & 1 & 0 & 0 \\
0&1&0&0\\
0&0&0&2\\
0&0&2&0
\end{matrix}
$$
This matrix has eigenvalues 1, 2, & -2, I've solved the eigenvectors for 1 & 2, but not for -2. Subtracting (-2) from the diagonal produces:
... | you can take advantage of the block structure of the matrix. you have two $2 \times 2$ matrices $B$ and $C$ where $B = \pmatrix{1&1\\0&1}, C = \pmatrix{0&2\\2&0}.$ $B$ has eigenvalues $1,1$ but only one eigenvector $(1,0)^T$ and $C$ has eigenvalues $-2, 2.$ the eigenvector corresponding to $-2$ is $(1,-1)^T$ and an e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1191247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proving $2(\sqrt{n} - 1) < \sum\limits_{i=1}^n\frac{1}{\sqrt i}$ I'm trying to prove $$2(\sqrt{n} - 1) < \sum_{i=1}^n\frac{1}{\sqrt i}$$ (Which is the opposite pretty much of Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$)
And I'm encountering some troubles I fail to see through.
My at... | This is just to illustrate a non-inductive method to proving the inequality:
\begin{align}\sum_{i = 1}^n \frac{1}{\sqrt{i}} &= 2\sum_{i = 1}^n \frac{1}{2\sqrt{i}} \\ & =2\sum_{i = 1}^n \frac{1}{\sqrt{i} + \sqrt{i}} \\
&> 2\sum_{i = 1}^n \frac{1}{\sqrt{i+1} + \sqrt{i}} \\
&= 2\sum_{i = 1}^n (\sqrt{i+1} - \sqrt{i})\\
&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1192428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $ Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $
Can someone please explain to me how to go about doing this?
| If
\begin{align*}
\cos\varphi & = \frac{b}{\sqrt{a^2 + b^2}}\\
\sin\varphi & = \frac{a}{\sqrt{a^2 + b^2}}
\end{align*}
then
$$a\sin\theta - b\cos\theta = -\sqrt{a^2 + b^2}\cos(\theta + \varphi)$$
since
\begin{align*}
-\sqrt{a^2 + b^2}\cos(\theta + \varphi) & = -\sqrt{a^2 + b^2}(\cos\theta\cos\varphi - \sin\theta\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1192729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Taylor series of $x/(x^2-4x+5)$ I'm supposed to find the Taylor series of this function (I can choose to center it at any A I want):
$$f(x)= x/(x^2-4x+5)$$
When I derivate, it only gets more and more confusing. How can I make any sense out of this?
| The first step is to complete the square: $x^2-4x+5 = (x-2)^2 + 1$. This suggests centering the expansion at $x-2$, since then the formula for the sum of a geometric series gives
$$
\frac{1}{1 + (x-2)^2} = \sum_{i=0}^\infty (-1)^i (x-2)^{2i}.
$$
Writing $x = (x-2) + 2$ we deduce
$$
\frac{x}{1 + (x-2)^2} = \sum_{i=0}^\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1193204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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The order of accuracy of the implicit Euler method is equal to $1$ I want to show that the order of accuracy of the implicit Euler method is equal to $1$.
That's what I have tried:
We have the initial value problem
$\left\{\begin{matrix}
y'(t)=f(t,y(t)) &, a \leq t \leq b \\
y(a)=y_0 &
\end{matrix}\right.$
Using Tayl... | Your first line should be, using $y(t^n)=y(t^{n+1})-h·f(t^{n+1}, y(t^{n+1}))+\frac{h^2}{2}·y''(\xi_n)$ as the appropriate Taylor expansion,
$$
|y(t^{n+1})-y^{n+1}|
=\left|y(t^n)-y^n+h·\bigl(f(t^{n+1},y(t^{n+1}))-f(t^{n+1},y^{n+1})\bigr)+ \frac{h^2}{2}·y''(\xi_n)\right|
\\
\leq |y(t^n)-y^n|+h·L·|y(t^{n+1})-y^{n+1}|+ \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1194443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find all prime numbers p such that both numbers $4p^2+1$ and $6p^2+1$ are prime numbers? I tried $p$ for $2, 3$ and $5$ and they are not primes for both cases. How can I find all these prime numbers that satisfy those conditions?
| You might want to double-check your numbers: $4 \times 5^2 + 1 = 4 \times 25 + 1 = 101$, which is prime, and $6 \times 5^2 + 1 = 6 \times 25 + 1 = 151$, which is also prime.
But you're right about $2$ and $3$: $4 \times 2^2 + 1 = 17$, which is prime, but $6 \times 2^2 + 1 = 25$ which is obviously composite; and $4 \tim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Various evaluations of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$ I recently ran into this series:
$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$$
Of course this is just a special case of the Beta Dirichlet Function , for $s=3$.
I had given the following solution:
$$\begin{aligned}
1-\frac{1}{3^3}+\frac... | Differentiating twice the logarithm of the Weierstrass representation of sine gives
$$ \sum\limits_{n=-\infty}^{\infty} {1\over (z+n)^2}=\frac{\pi^{2}}{\sin^{2}(\pi z)} $$
(as i've been answered in here.)
Now differentiate once more and consider $z=\frac{1}{4}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
Find solution of $(1-x^2)y''-xy'+p^2y=0, p \in \mathbb{R}$ The following differential equation is given:
$$(1-x^2)y''-xy'+p^2y=0, p \in \mathbb{R}$$
*
*Find the general solution of the differential equation at the interval $(-1,1)$ (with the method of power series).
*Are there solutions of the differential equation... | Those ratios simplify rather nicely, so you can indeed deduce something about the radius of convergence. But in fact there is a general theory that says there should be analytic solutions in any region (in the complex plane) where the coefficients are analytic and the leading coefficient is nonzero. In this case the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1197701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finf $f(x)$ which is a second degree polynomial, such that $f(1)=0$ and $f(x) = f(x-1)$ I must find a function $f(x) = ax^2+bx+c$ such that:
$$f(1) = a+b+c=0\\f(x)=f(x-1)\implies ax^2+bx+c = a(x-1)^2+b(x-1)+c\implies\\ax^2+bx+c = ax^2+(-2a+b)x+a-b+c\implies\\a = a, b = -2a+b, c = a-b+c$$
but this results fo $a=b=c=0$. ... | You simply get that
$$\{f(x)=ax^2+bx+c\mid f(1)=0, f(x)=f(x-1)\}=\{0\}$$
which is actually correct.
But in other way, you can say that $$\{f(x)\in \mathbb R_2[x]\mid \deg f=2,\ f(1)=0,\ f(x)=f(x-1)\}=\emptyset$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
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Finding the limit $\lim_{x\rightarrow \infty} \sqrt[3]{x+1}-\sqrt[3]{x}$ I am trying to find this limit
$$\lim_{x\rightarrow \infty} \sqrt[3]{x+1}-\sqrt[3]{x}$$
My so far method is this
*
*$f(x)>0.$
*$f^{\prime}(x)=\frac{1}{3\sqrt[3]{(x+1)^2}}-\frac{1}{3\sqrt[3]{x^2}}<0.$
*For every $0<y<1$ the equation $f(x)=y$ ... | Hint:-
$$0<\left(\sqrt[3]{x+1}-\sqrt[3]{x}\right)=\displaystyle\frac{1}{\left(\sqrt[3]{(x+1)^2}+\sqrt[3]{x(x+1)}+\sqrt[3]{x^2}\right)}<\displaystyle\frac{1}{3\sqrt[3]{x^2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 4
} |
Solve this system of equation Solve this system of equations for real $x$ and $y$:
*
*$5x\left(1+\dfrac{1}{x^2+y^2}\right)=12$
*$5y\left(1-\dfrac{1}{x^2+y^2}\right)= 4$
I juggled with those equations and got $x-y+\dfrac{x+y}{x^2+y^2}=\dfrac{8}{5}$, from where I guessed a solution $(2,1)$.
But I don't know how to ... | $$\left\{ {\begin{array}{*{20}{c}}{5x\left( {1 + \frac{1}{{{x^2} + {y^2}}}} \right) = 12}\\{5y\left( {1 - \frac{1}{{{x^2} + {y^2}}}} \right) = 4}\end{array}} \right.$$
If you sum the above equations, you can find formula of a line all the points on it are solutions to that system of equations;
$$\left\{ {\begin{array}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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How come, in this problem, the maximum product is always achieved using only $2$s and $3$s? Consider the following problem.
Given a number $N$, write it as a sum $n = n_1 + n_2 + \cdots + n_k$, such that the product $p = n_1 \times n_2 \times \cdots \times n_k$ is maximized.
For example, $11$ can be written as $2 + 3 +... | Any factor $1$ can be merged with some other factor, and the product will get larger. When $n>4$ then $(n-3)\cdot 3>n$, furthermore $4=2\cdot2$. Therefore the largest possible product can be written with $3$s and $2$s only.
As $2\cdot 2\cdot 2<3\cdot 3$ it is advantageous to replace three $2$s by two $3$s as often as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that the algebraic expressions are equivalent.
$$\frac{3^{k+1}-1}{2} + 3^{k+1} = \frac{3^{k+2}-1}{2}$$
with steps make left hand side = right hand side by modifying one or both expressions
Thanks for your help guys, I solved it like this:
$$\frac{3^{k+1}-1}{2} + 3^{k+1} = \frac{3^{k+1}-1}{2} + \frac{2*3^{k+1}}... | $${3^{k+1}-1\over 2}+3^{k+1}={3^{k+1}\over 2}-{1\over 2}+{2(3^{k+1})\over 2}=({3\over 2})3^{k+1}-{1\over 2}={3^{k+2}-1\over 2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Find integers $a,b,c,d$ such that is it possible for someone to run a computer search for me to turn up integers $a,b,c,d$ such that $$1+\sqrt{2}+\sqrt{3}+\sqrt{6}=\sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d}}}}$$? Note: I know they exist, I just can't find them.
| Problem statement:
$$1+\sqrt{2}+\sqrt{3}+\sqrt{6}=\sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d}}}}$$
Let's try this identity that you posted:
$$\sqrt{x}+\sqrt{y}=\sqrt{x+y+\sqrt{4xy}}$$
Use your variables:
$$\sqrt{3}+\sqrt{6}$$
$$=\sqrt{3+6+\sqrt{4\cdot 3\cdot 6}}$$
$$=\sqrt{9+\sqrt{72}}$$
$$\sqrt{1}+\sqrt{2}$$
$$=\sqrt{1+2+\sqrt{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, find the first 6 terms in the Taylor expansion solution $y=\varphi (x)$ Given $\frac{dy}{dx}=x^2+y^2$ and initial condition $\varphi (0)=1$, use the method of reduction to an integral equation and successive approximation to find the first 6 terms in ... | $$y = 1 + \int_0^x y^2 + t^2 \, dt$$ we will define $$y_{n+1} = 1 + \int_0^x (t^2 + y_n^2)\, dt = 1 + \frac 13 x^3 + \int_0^x y_n^2 \, dt,\, y_0 = 1.$$ so $$y_1 = 1+ \frac 13 x^3 + \int_0^x \,dt = 1 + x + \frac 13 x^3 \tag 1\\ y_2 = 1 + \frac 13 x^3 + \int_0^x \left(1 + t + \frac 13 t^3\right)^2\, dt = 1 + \frac 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1205397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove the logarithmic inequality Prove that:
$(\log_{24}{48})^2+(\log_{12}{54})^2>4$
I tried to put $t=\log_23$ and get the equation $6t^4+32t^3+22t^2-84t-74>0$.
But I can't do anything with it...
| Let $l=\log_{2}3$, so $\log_{3}2=\frac{1}{l}$ and $l<\frac{8}{5}$ since $2^8>3^5$, and $l>\frac{11}{7}$ since $2^{11}<3^7$.
Then $\displaystyle\log_{24}48=\frac{\log_{2}48}{\log_{2}24}=\frac{4+l}{3+l}>\frac{4+8/5}{3+8/5}=\frac{28}{23}\;\;\;$ and $\hspace{.36 in}\displaystyle\log_{12}54=\frac{\log_{3}54}{\log_{3}12}=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1206452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find a vector whose image under $T$ is the vector $b$
Hey everyone, I'm having some trouble solving this problem. To find the vector can I multiply the matrix $A$ by the column vector $[a, b, c]$ and set that equal to the vector $b$? or do I have to row reduce to solve this problem? Thanks for the help
| An alternative way, using advanced methods: Scalar product and cross product.
Setting the columns from $A$
$$u = \left( {\begin{array}{*{20}{c}}
4 \\
{ - 3} \\
5
\end{array}} \right),v = \left( {\begin{array}{*{20}{c}}
4 \\
{ - 6} \\
{ - 3}
\end{array}} \right),w = \left( {\begin{array}{*{20}{c}}
3 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1209197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Elementary proof that $\sum_{n \geq 1} \frac{(-1)^n}{\sqrt{n}}+i \frac{1}{n^2}$ is conditionally convergent. I must prove that the complex series $$\sum_{n \geq 1} \frac{(-1)^n}{\sqrt{n}}+i \frac{1}{n^2}$$ is conditionally convergent. The catch is, I'm not supposed to use anything "too advanced", meaning I only have in... | Since $$\sqrt{\frac{1}{n} + \frac{1}{n^4}} \ge \sqrt{\frac{1}{n}} = \frac{1}{\sqrt{n}}$$ and $\sum_{n = 1}^\infty \frac{1}{\sqrt{n}}$ diverges, by direct comparison the series
$$\sum_{n = 1}^\infty \left|\frac{(-1)^n}{\sqrt{n}} + \frac{i}{n^2}\right| = \sum_{n = 1}^\infty\sqrt{\frac{1}{n} + \frac{1}{n^4}}$$ diverges.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1209397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Help to resolve a Double Integral I'm doing a workout guide about double integrals and I came across an exercise that I could not resolve for a while.
$$\int_0^2\int_1^2 \frac{x}{\sqrt{1+x^2+y^2}} \,\mathrm dx\,\mathrm dy$$
I guess that the easier order of integration is $dxdy$, because if I try to integrate respect to... | The antiderivative may be deduced, but it takes a little patience. I prefer to integrate by parts rather than do a trig sub right away.
$$\begin{align}\int dx \sqrt{x^2+a^2} &= x \sqrt{x^2+a^2} - \int dx \frac{x^2}{\sqrt{x^2+a^2}}\\ &= x \sqrt{x^2+a^2} - \int dx \sqrt{x^2+a^2} + a^2 \int \frac{dx}{\sqrt{x^2+a^2}} \end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1209636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Let $a,b,c$ be the nonnegative real numbers such that $a+b+c=1$. Prove that $\sqrt{a+\frac{(b-c)^2}4}+\sqrt b+\sqrt c\le\sqrt3$
Let $a,b,c$ be the nonnegative real numbers such that $a+b+c=1$. Prove that
$$\sqrt{a+\frac{(b-c)^2}4}+\sqrt b+\sqrt c\le\sqrt3$$
I first wrote $a$ as $1-b-c$ and substituted it in main in... | I use Cauchy-Schwarz inequality we have
$$\left(\sqrt{a+\dfrac{(b-c)^2}{4}}+\sqrt{b}+\sqrt{c}\right)^2
\le \left(a+\dfrac{(b-c)^2}{4}+\dfrac{(\sqrt{b}+\sqrt{c})^2}{2}\right)(1+2)
$$
$$\Longleftrightarrow \left(a+\dfrac{(b-c)^2}{4}+\dfrac{(\sqrt{b}+\sqrt{c})^2}{2}\right)\le 1$$
since $1-a=b+c$
$$\Longleftrightarrow \dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1209896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Sum of Fibonacci numbers While trying to find find a formula to calculate the length of the golden spiral I came across the sum of the Fibonacci numbers.
I noticed that
$$\text{Fibonacci numbers: }1,1,2,3,5,8,13,21,34...$$
$$1+1+2= 5-1$$
$$1+1+2+3= 8-1$$
and that
$$2+3+5+= 13-2$$
$$3+5+8=21-5$$
so generalized that writ... | Fibonacci numbers can be written as a matrix using:
$$\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n = \begin{bmatrix} F_{n+1} & F_{n} \\ F_{n} & F_{n-1}\end{bmatrix}$$
So that any sum, using $X= \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$, is :
$$\sum_{k=a}^b F_n = \left( \sum_{k=a}^b X^n \right)_{2,1}$$
which is a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1211909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Probability that minimum of two numbers is less than 4 Suppose I have to choose two numbers from set $$S=\{1,2,3,4,5,6 \}$$ without a replacement , then what is the probability that minimum of two is less than $4$?
I made two groups for this problem $A= \{1,2,3 \}$ and $B=\{4,5,6 \}$
.There are two possibilities , eith... | The error with your approach is that "One from $A$ and one from $B$" has two ways of happening: $A$ then $B$, or $B$ then $A$. So you need to multiply your calculation of this case by 2.
$$P(E) = \frac{3}{6} \cdot \frac{2}{5} + 2 \cdot \frac{3}{6} \cdot \frac{3}{5} = \frac{4}{5}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1213148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Lowering powers of $\cos^2x \sin^4x$ First, I will be straight up, this is a homework question.
I need to write $\cos^2 x \sin^4 x$ in terms of cosine to the first power. I know that $\sin^4x$ =
$$ \frac{3-4\cos 2x+\cos 4x}{8}$$
from there I go:
$$ \frac{1+\cos 2x}{2} \cdot \frac{3-4\cos 2x+\cos 4x}{8}$$
$$ \frac{3(1 ... | There is a systematic approach to solve such questions, based on complex numbers.
We have $\cos kx=\dfrac{e^{ikx}+e^{-ikx}}2=\dfrac{z^k+z^{-k}}2$ and $\sin kx=\dfrac{e^{ikx}-e^{-ikx}}{2i}=\dfrac{z^k-z^{-k}}{2i}$, where $z=e^{ix}$.
The expression can be rewritten
$$\left(\frac{z+z^{-1}}{2}\right)^2\left(\frac{z-z^{-1}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1216162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Proving $\frac{1}{\sqrt{2}}=1-\frac{1}{2^2}-\frac{1}{2!2^4}-\frac{3!!}{3!2^6}-\frac{5!!}{4!2^8}-\cdots$ How can I prove
$$\frac{1}{\sqrt{2}}=1-\frac{1}{2^2}-\frac{1}{2!2^4}-\frac{3!!}{3!2^6}-\frac{5!!}{4!2^8}-\cdots$$
I wanted to prove it by using the Taylor series of $\sqrt{2}$, but I couldnt do.
| As Lucian pointed out, we can write
$\displaystyle\frac{1}{\sqrt{2}}=\left(1-\frac{1}{2}\right)^{1/2}=\sum_{k=0}^{\infty}\binom{1/2}{k}\left(-\frac{1}{2}\right)^k=1+\sum_{k=1}^{\infty}\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)\cdots(\frac{1}{2}-(k-1))}{k!}\cdot\frac{(-1)^k}{2^k}$
$\displaystyle=1+\sum_{k=1}^{\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1217847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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find taylor series to fourth term I'm wondering if there is faster method than just calculating derivatives with finding taylor series up to 4 term of function $\displaystyle f(x)=\frac{(1+x^4)}{(1+2x)^3(1-2x)^2}$
| The idea is to use the formula
$$
\frac{1}{(1-x)^t} = \sum_{n=0}^\infty \binom{n+t-1}{n} x^n,
$$
which can be proved by induction on $t$.
Using this we get
$$
\frac{1+x^4}{(1+2x)^3(1-2x)^2} = \\
(1+O(x^4))(1 - 6x + 24x^2 - 80x^3+O(x^4))(1 + 4x + 12x^2 + 32x^3 + O(x^4)) = \\
1 + (-6+4)x + (24-6\cdot 4+12)x^2 + (-80 + 24... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1219248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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How to differentiate $y=(x+1)^3/x^{3/2}$ and $y=2x^4/(b^2-x^2)$ I need to solve a list of derivatives to help me on an exam; however, I'm in doubt when they use another variable (constant) or when I have a fraction with functions that use the power rule.
For example:
$$y = \frac{(x + 1)^3}{x^{3/2}}.$$
This one I tried ... | When the expression contains products and ratioé, logarithmic differentiation make life much easier.
Let us consider the first problem $$y = \frac{(x + 1)^3}{x^{3/2}}$$ Taking logarithms $$\log(y)=3\log(x+1)-\frac32\log(x)$$ Now, differentiate both sides $$\frac{y'}y=\frac3{x+1}-\frac3{2x}=\frac{3 (x-1)}{2 x (x+1)}$$ $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1219363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Definite integral with trigonometric functions I have problem finding, how to solve this integral
$$
\int _0 ^{\frac{\pi}{4}} \frac{3 \sin x + 2 \cos x}{2 \sin x + 3 \cos x}dx
$$
This I can rewrite as
$$
\int _0 ^{\frac{\pi}{4}} \frac{12 \sin ^2 x - 5 \cos x \sin x - 6}{4-13 \cos ^2 x}dx =
$$
$$
= \int _0 ^{\frac{\pi}{... | Whenever you have to integrate a fraction involving odd powers of $\sin$ and $\cos$, the preferred algorithmic approach is the substitution $t = \tan \frac x 2$. This leads to $\mathrm{d}x = \frac 2 {1 + t^2} \mathrm{d}t$ and then, using that $\frac {\sin x} {\cos x} = \tan x = \tan (\frac x 2 + \frac x 2) = \frac {2t}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1222861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to deduce that $1\cdot 1 + 2\cdot 1 + 2\cdot 2 + 3\cdot 1+3\cdot 2+3\cdot 3 +...+(n\cdot n) = n(n+1)(n+2)(3n+1)/24$ I know how to reason $$1\cdot2 + 2\cdot3 + 3\cdot4 + n(n-1) = \frac{1}{3}n(n-1)(n+1)$$
However, I'm stuck on proving $$1\cdot1 + (2\cdot1 + 2\cdot2) + (3\cdot1+3\cdot2+3\cdot3) + \cdots +(n\cdot 1+...... | You can do as the following :$$\begin{align}\sum_{k=1}^{n}k(1+2+\cdots+k)&=\sum_{k=1}^{n}k\cdot\frac{k(k+1)}{2}\\&=\frac 12\sum_{k=1}^{n}k^3+\frac 12\sum_{k=1}^{n}k^2\\&=\frac 12\left(\frac{n(n+1)}{2}\right)^2+\frac 12\cdot\frac{n(n+1)(2n+1)}{6}.\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1225311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Trig function integral I'm trying to solve
$$\int_{0}^{\pi}\frac{dx}{cos^2(x)-a^2}, \hspace{5mm} 0<a<1$$
There are numerous examples of similar integrals but non with the condiction that $0<a<1$, say $a = 0.5$. Since the function is even one can expand the domain of integration to $2\pi$ and use the residuum theorem, b... | The Cauchy principal value of the integral is $0$.
We have
$$F(a) = 2\int_0^{\pi/2} \dfrac{dx}{\cos^2(x)-a^2} = 2 \int_0^{\pi/2} \dfrac{\sec^2(x)dx}{1-a^2\sec^2(x)}$$
Setting $\tan(x) = t$, we obtain
\begin{align}
F(a) & = 2 \int_0^{\infty} \dfrac{dt}{1-a^2-a^2t^2} = \dfrac2{a^2} \int_0^{\infty} \dfrac{dt}{\left(\dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1227237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Number in tens place Can you figure out the digit in the tens place of the solution to $4^{2015} \cdot 9^{2016}$ without using a calculator?
By tens place I mean, for example if you have number $2451$, the number in tens place here is $5$.
I know the answer, but I don't know how to get it, so if you got any ideas, plea... | You have
$$4^{11}=4194304\equiv 4 \pmod{100}$$
And since $2015=183\times 11+2$, and $183=16\times 11+7$, you have
$$4^{2015}=4^{11\times183+2}=4^2\times(4^{11})^{183}\equiv 4^2\times 4^{183} \pmod{100}$$
$$=4^2\times4^{11\times16+7}=4^9\times(4^{11})^{16}\equiv 4^9\times4^{16}=4^{25}=4^3\times(4^{11})^2\equiv4^5 \pmod{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1229260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
limit using taylor series I keep getting an error in the expansion
$$\lim_{x\to 0 }\frac{2\exp(\sin(x))-2-x-x^2-\arctan (x) }{x^3}$$
The numerator works out as
$$\approx 2 (1+\sin(x) + 1/2 \sin^2(x)) -2-x-x^2 - (x-x^3/3+x^5/5)$$
$$\approx 2(1+x-x^3/3!+1/2(x-x^3/3!)^2 -2-2x+x^3/3-x^2 -x^5/5 $$
$$ = o(x^4)$$
so that the ... | Starting from $$e^y=1+y+\frac{y^2}{2}+\frac{y^3}{6}+\frac{y^4}{24}+O\left(y^5\right)$$ Replace $y$ by the expansion of $\sin(x)$ to get $$e^{\sin(x)}=1+x+\frac{x^2}{2}-\frac{x^4}{8}-\frac{x^5}{15}+O\left(x^6\right)$$ Just as quid answered while I was typing, one term was missing in the expansion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1229968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Simplify $ \frac{b}{a-b}+ \frac{a}{b-a}$ I have a question regarding simplification of an algebraic expression.
Here is the problem:
$ \frac{b}{ a - b} + \frac{a}{b-a} $
The outcome is $ -1 $
Here is how I try to simplify it:
*
*Add fractions: $ \frac{ b(-a+b) + a(a-b)}{(a-b)(-a+b)} $
*FOIL: $ \frac{a^2 - 2ab + b... | Notice that $-a+b=-(a-b)$. Thus: $\frac{a-b}{-a+b}=\frac{a-b}{-(a-b)}=-\frac{a-b}{a-b}=-1$.
You could also have done this in a much easier way:
$\frac{b}{a-b}+\frac{a}{b-a}=\frac{b}{a-b}-\frac{a}{-(b-a)}=\frac{b}{a-b}-\frac{a}{a-b}=\frac{b-a}{a-b}=-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1231556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Prove the following trigonometric identity $$\frac{\tan{(\frac{\pi}{4}+x)}-\tan{(\frac{\pi}{4}-x)}}{\tan{(\frac{\pi}{4}+x)}+\tan{(\frac{\pi}{4}-x)}} = 2\sin{x}\cos{x}$$
==============
On L.H.S, I've tried to write it using the sum and difference formula so it becomes
$$\frac{\dfrac{1+\tan x}{1-\tan x}-\dfrac{1-\tan x}{... | You may write
$$
\begin{align}
\dfrac{\dfrac{1+\tan x}{1-\tan x}-\dfrac{1-\tan x}{1+\tan x}}{\dfrac{1+\tan x}{1-\tan x}+\dfrac{1-\tan x}{1+\tan x}}=\dfrac{(1+\tan x)^2-(1-\tan x)^2}{(1+\tan x)^2+(1-\tan x)^2}
&=\dfrac{4\tan x}{2(1+\tan^2x)}\\\\&=2\tan x \cos^2x\\\\&=2\sin x\cos x
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1232129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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If the distance between two lines is $ \frac{1}{\sqrt{3}} $, then $ \alpha $ is...
If the shortest distance between the lines $\displaystyle \frac{x-1}{\alpha}=\frac{y+1}{-1}=\frac{z}{1}\;, (\alpha \neq - 1)$ and
$x+y+z+1=0 = 2x-y+z+3 = 0$ is $\displaystyle \frac{1}{\sqrt{3}}\;,$ Then $\alpha = $
$\bf{My\; Solution::... | I like to use Plücker coordinates for lines
*
*Line along $(\alpha,-1,1)$, through Point $(1,-1,0)$
$$ L_1 = \left\{ \begin{array}{c} \vec{e} \\ \vec{r}\times\vec{e} \end{array} \right\} = \left\{ \begin{array}{c} \begin{pmatrix} \alpha\\-1\\1\end{pmatrix} \\
\begin{pmatrix} 1\\-1\\0\end{pmatrix} \times \begin{pm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1232568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Trigonometric integrals How do I evaluate this indefinite integral ? Integral $$\int\frac{x^2+n(n-1)}{(x\sin(x)+n\cos(x))^2}dx$$ What type of integral is it ? Is there any intuition involved in the approach to solve it?
Edit: The complete term in denominator has 2 as exponent. There was a typo previously.
| $\bf{Another\; Solution::}$ Given $$\displaystyle \int\frac{x^2+n(n-1)}{\left(x\sin x+n\cos x\right)^2}dx\;,$$ Now Multiply both $\bf{N_{r}}$ and $\bf{D_{r}}$
by $x^{2n-2};,$ We Get $$\displaystyle \int\frac{\left[x^{2}+n(n-1)\right]\cdot x^{2n-2}}{\left(x^{n}\sin x+nx^{n-1}\cos x\right)^2}dx$$
Now Let $$x^n\sin x+nx^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1233433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Find the largest $k$ such that $3^k$ divides the product of the first $100$ odd integers
Let $P$ be the product of the first 100 positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k$.
There are $50$ odd numbers and $50$ even numbers between $0$ and $100$, $99$ being the $50$th odd. S... | evidently the highest power of $3$ dividing the first $100$ odd numbers is the same as the highest power of $3$ which divides $\frac{200!}{100!}$
$$
200 = 2.3^4 +1.3^3 + 1.3^2+ 0.3^1+2.3^0
$$
so the sum-of-digits in the 3-ary represention of 200 is 6. similarly for 100 the sum of 3-ary digits is 4.
hence the number re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1234316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Find all solutions to $4x^2+6x+1 \equiv 0 \pmod {13}$ Find all solutions to $4x^2+6x+1 \equiv 0 \pmod {13}$
I think it has no solutions but I am not sure how to show this.
| We have $4x^2+6x+1 = (2x+3/2)^2-5/4$. Hence,
\begin{align}
4x^2+6x+1 \equiv 0 \pmod{13} \implies (2x+3/2)^2-5/4 \equiv 0 \pmod{13}
\end{align}
This simplifies into
\begin{align}
(4x+3)^2 \equiv 5 \pmod{13}
\end{align}
This has no solutions, since
$$5^{(13-1)/2}\pmod{13} \equiv 5^6\pmod{13} \equiv (25)^3 \pmod{13} \equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1235156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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What is that function? Polynomial? Is it a polynomial or rational polynomial or else?
$y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$
I need to fit a curve to a discrete data of that form, so I need to know what fitting to use.
| We have $$\frac{a}{x^4}+\frac{b}{x^2}+c=\frac{a}{x^4}+\frac{bx^2}{x^4}+\frac{cx^4}{x^4}=\frac{a+bx^2+cx^4}{x^4}$$
This is a rational function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1239102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove by mathematical induction: $\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{n^2}>1$ Could anybody help me by checking this solution and maybe giving me a cleaner one.
Prove by mathematical induction:
$$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{n^2}>1; n\geq2$$.
So after I check special cases for $n=2,3$, I have to prove... | Your argument is fine and quite clearly presented. You can shorten the presentation considerably, though, by doing something like this:
For $n\ge 2$ let $a_n=\sum_{k=n}^{n^2}\frac1k$. Since $a_2=\frac12+\frac13+\frac14>1$, it suffices to show that $a_{n+1}\ge a_n$ for $n\ge 2$. Since $n^2+2n+1<2n^2+n$ for $n\ge 2$, we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1239518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 0
} |
Antiderivative of $ (x^2 + c)^{-3/2} $ What method should be used to determine the antiderivative of this expression?
Edit: I have $ c > 0 $ in the problem I'm working on.
| $$\int(x^2+c)^{-3/2}dx=\int\frac{1}{\sqrt{(x^2+c)^3}}$$
Let us use a new variable $t$ such that $\sqrt{x^2+c}=x+t$. This is the first Euler substitution.
From this formula we can solve for $x$ to get $x=\frac{c-t^2}{2t}$, $dx=\frac{-c-t^2}{2t^2}dt$, $\sqrt{x^2+c}=x+t=\frac{c+t^2}{2t}$, and $\sqrt{(x^2+c)^3}=\frac{(c+t^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Fermat numbers of the form of $b^2$ For n > 1 Let $F_n = 2^{2^n} + 1$ be a fermat number and b = $2^{2^{n - 2}}$ * ($2^{2^{n - 1}}$ - 1 ).
Then $b^2$ $\equiv$ 2 (mod $F_n$)
I tried to square the original expression I got something ugly that I couldn't simplify further.
I got $b^2$ = $2^{2^{n - 1}}$ * ($2^{2^n}$ - $2 * ... | You're almost there! Just notice that $2^{2^n}\equiv -1\pmod{2^{2^n}+1}$ :
$$\begin{align}b^2&=2^{2^{n-1}}(2^{2^n}-2*2^{2^{n-1}}+1)\\&=2^{2^{n-1}}(2^{2^n}+1-2*2^{2^{n-1}})\\&\equiv2^{2^{n-1}}(0-2*2^{2^{n-1}})\pmod{2^{2^n}+1}\\&\equiv-2*2^{2^{n-1}}*2^{2^{n-1}}\pmod{2^{2^n}+1}\\&\equiv-2*2^{2^{n-1}+2^{n-1}}\pmod{2^{2^n}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Computing $\int_3^5 \frac{x^2\,dx}{\sqrt{(x-3)(5-x)}}$
$$
\int_3^5 \frac{x^2\,dx}{\sqrt{(x-3)(5-x)}}
$$
how? $x^2/\sqrt{8x-x^2-15}$ and what to do then?
| $$
-(x^2 - 8x + 15) = 1-(x^2 - 8x + 16) = 1 -(x-4)^2 = 1-\sin^2\theta
$$
So the square root of this is $\cos\theta$.
$$
dx = \cos\theta\,d\theta
$$
$$
x^2 = (4+\sin\theta)^2
$$
As $x$ goes from $3$ to $5$, $\sin\theta=x-4$ goes from $-1$ to $1$, so $\theta$ goes from $-\pi/2$ to $\pi/2$. We get
$$
\int_{-\pi/2}^{\pi/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How to evaluate volume of oblique frustum of right circular cone with elliptical section?
Let there be an oblique frustum, with an elliptical section, of a right circular cone with apex point O & cone angle $2\alpha=60^{o}$. It is obtained by cutting the cone by a plane at a normal distance $OM=h=20 cm$ & making an an... | First, assign an $xyz$ coordinate system. For simplicity, let $M$ be the origin, and let the positive $x$-axis point to the right, and the positive $y$-axis point into the screen, while the $z$-axis points upward.
Next, find the unit vector in this coordinate system of the axis of the cone. This can be found quite ea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Solve for $x:1 + \tan^2(x) = 8\sin^2(x)$ I have a tricky problem , I tried several methods and I can't seem to get a definite answer.
$1 + \tan^2(x) = 8\sin^2(x), x \in [\frac{\pi}{6} , \frac{\pi}{2}]$
I got to $8\cos^4(x)-8\cos^2(x)+1=0$ and found that $\cos^2(x) = \frac{1}{4}[2-\sqrt{2}]$ but that is not too useful... | Left side is $ \sec^2 x $ and exploiting $2\sin x \cos x =\sin 2x $ should ring a bell.
$$2\sin^2 2 x = 1 \implies \sin 2x = \pm \dfrac1{\sqrt2} $$
where $ x = \frac12 $ of $\pi/4 = \pi/8 $ in the first quadrant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
} |
"Simpler" geometrical description So i was asked to find:
Find the matrix that represents the linear transformation of the plane obtained by:
*
*reflecting in the line y = x, $\begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}$
*then rotating anticlockwise through an angle of 45 degrees, $\begin{bmatrix} \frac{1}{\sqrt{2}}&-... | there is a result from geometry that says any rotation by angle $2\theta$ is equivalent to two reflections on mirrors separated by an angle $\theta.$ the two mirrors can be placed on ant line through the point of rotation.
we will also use the fact that reflections are involuntary; that they are their own inverses.
l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$a^2 = 2b^3 = 3c^5$ Find the smallest value of $abc$. We have following equation:
$a^2 = 2b^3 = 3c^5$
Where $a, b, c$ are natural numbers.
Find the smallest possible value of product $abc$.
| Since $6\mid c$ we may write $c=2^r3^s$ with $r,s\ge 1$, so that $a^2=3c^5=2^{5r}3^{5s+1}$. Since $a^2$ is a square, $5r$ and $5s+1$ must be even. The minimal $r,s$ are then $r=2$ and $s=1$, so that $c=12$ and $a=864$. It follows that $b=72$, and
$$
abc=a^2=3b^3=2c^5.
$$
This value is minimal, because $c$ is minimal (s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Am I misinterpreting this matrix determinant property? I was reading matrix determinant properties from wikipedia.
The property reads
$\det(cA) = c^n \det(A)$ for $n \times n$ matrix.
However I am not able to realize it. What I find is $\det(cA) = c\det(A)$
For example, multiplying matrix by 2 and then taking the dete... | You have $\det (cB) = \det (cI) \det B $ and you can see from the formula for $\det$ that $\det (cI) = c^n$.
Another way is to notice that $\det$ is a multilinear function of the columns (or rows), that is, we can write
$\det(A) = f(a_1,...,a_n)$ where $a_k$ is the $k$th row of $A$, and $f$ is
linear in each argument s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1250186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
In how many ways can $1000000$ be expressed as a product of five distinct positive integers? I'm trying to solve the following problem:
"In how many ways can the number $1000000$ be expressed as a product of five distinct positive integers?"
Here is my attempt:
Since $1000000 = 2^6 \cdot 5^6$, each of its divisors has ... | For $n{\ge}0$, $\;$ let $F(n)$ be the ways in which the number $10^n$ can be expressed as a product of five distinct positive integers.
Then
$$F(n)=\left(\frac{1}{5!}\right)\left(24\left(\left\lfloor\frac{n}{5}\right\rfloor-\left\lfloor\frac{n-1}{5}\right\rfloor\right)^{2}
+(-30)\left(\left\lfloor\frac{n}{4}\right\rflo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1251256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 2
} |
Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$ When I was playing with numbers, I found that there are many triplets of three positive integers $(a,b,c)$ such that
*
*$\color{red}{2\le} a\le b\le c$
*$\sqrt{abc}\in\mathbb N$
*$\sqrt{abc}$ divides $(a-1)(b-1)(c-1)$
Exampl... | We study solutions, if any, of the shape $(a,b,c) = (x^2 P + 1,~ y^2 P + 1,~ z^2)$. We need
$$z^2 \left(x^2 y^2 P^2 + (x^2 + y^2) P + 1 ) \right)$$
be a square. Namely, we need
\begin{align}
2 x^2 y^2 P &= -(x^2 + y^2) + \sqrt{ (x^2 + y^2)^2 + 4(w^2 - 1)x^2 y^2} \\
\implies P &= \frac{ -(x^2 + y^2) + u }{ 2 x^2 y^2}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1251576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "56",
"answer_count": 3,
"answer_id": 2
} |
Finding a matrix from equation we've got the following 4x4 Matrix
$$\begin{pmatrix}
4 & -2 & 3 & 2\\
3 & 5 & 1 & -4\\
-1 & 6 & -4 & -7\\
-2 & 0 & -2 & 4
\end{pmatrix}$$
and I need to find $B$ from the equation: $(A-3I)B=0$.
i started to solve it by finding first $A-3I$. and I got:
$$\begin{pmatrix}
1 & -2 & 3 & 2\\
... | $$\begin{pmatrix}
1 & -2 & 3 & 2\\
3 & 2 & 1 & -4\\
-1 & 6 & -7 & -7\\
-2 & 0 & -2 & 1
\end{pmatrix}$$
is a zero divisor since : column$(3)=$column$(1)-$column$(2)$.
So the matrix:
$$B=\begin{pmatrix}
0 & 0 & -a & 0\\
0 & 0 & a & 0\\
0 & 0 & a & 0\\
0 & 0 & 0 & 0
\end{pmatrix}$$
gives $AB=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Poisson Integral relation If $$ I_n(r) = \int_0^\pi \frac{\cos nx}{r^2-2r\cos x+1} \, dx $$
How to prove that
$$ I_{n-1}(r)+I_{n+1}(r)= \left(r+\frac{1}{r}\right)I_n(r)\text{ ?}$$
I only find that $$I_{n-1}(r)+I_{n+1}(r)= \int_0^\pi \frac{2\cos nx\cos x}{r^2-2r\cos x+1} \, dx$$
| $$\begin{aligned}
I_{n-1}(r)+I_{n+1}(r) &=\int_0^{\pi}\frac{2\cos(nx) \cos x}{r^2-2r\cos x+1}\,dx \\
&=-\frac{1}{r}\int_0^{\pi} \frac{\cos(nx)(r^2-2r\cos x+1-r^2-1)}{r^2-2r\cos x+1}\,dx \\
&=-\frac{1}{r}\int_0^{\pi}\cos(nx)\,dx+\frac{r^2+1}{r}\int_0^{\pi} \frac{\cos nx}{r^2-2r\cos x+1}\,dx\\
&=\left(r+\frac{1}{r}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Factorize Trigonometric Equation: $ 3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0 $ I have a problem with the following trigonometric equation:
$$ 3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0 $$
It's from the book Engineering Mathematics 7th edition by Stroud.
The book is giving the answer, but I can't seem to be able t... | If you put $\sin x = a$, and $\cos x = b$, then you might be able to see the "structure" of the equation:
$$\begin{align} 3\sin^2x - 2\sin x \cos x - \cos^2x & = 3a^2 - 2ab - b^2 \\
&= 3a^2-3ab+ab-b^2\\ & =3a(a-b)+b(a-b)\\ & =(a-b)(3a+b)\\ & = (3a+ b)(a-b) \\ & = (3\sin x + \cos x)(\sin x - \cos x) = 0\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
How can a binomial coefficient can be approximated by using Stirling's formula? I've met some difficulties with such question:
How can we approximate a binomial coefficient by using a Stirling's factorial approximation.
I've evaluate a little bit and got this
How can I transform the right side of this equation for gett... | Another case is when
$k$ is a constant times $n$.
Let
$k = a n $
where $0 < a < 1$,
so $k/n = a$.
Then
$\begin{align}
\binom{n}{k}
&=\frac{n!}{k!(n-k)!}\\\\
&\approx. \frac{\sqrt{2\pi n}(\frac{n}{e})^n} {\sqrt{2\pi k}(\frac{k}{e})^k \sqrt{2\pi (n-k)}(\frac{n-k}{e})^{n-k}}\\\\
&=\frac{1}{\sqrt{2\pi k}(k/n)^k \sqrt{1-(k/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Diophantine equation $x^2 + xy + y^2 = \left({{x+y}\over{3}} + 1\right)^3$. Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
| Setting $x+y=3t$ and $xy=s$, we obtain that
$$9t^2-s = (t+1)^3 \implies s = -t^3+6t^2-3t-1$$
$x$ and $y$ satisfying the quadratic $a^2 -3ta + s =0$. This means $9t^2-4s = k^2$. Eliminating $s$, we obtain
$$4t^3-15t^2+12t+4 = k^2 \implies 64t^3 - 240t^2 + 192t + 64 = (4k)^2$$
$$(4t-5)^3 - 108t + 189 = (8k)^2 \implies (4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Prove that $n^2(n^2+1)(n^2-1)$ is a multiple of $5$ for any integer $n$. Prove that $n^2(n^2+1)(n^2-1)$ is a multiple of $5$ for any integer $n$.
I was thinking of using induction, but wasn't really sure how to do it.
| If one wishes to use induction:
The statement holds for $n=0$, so let us assume it's true for $n=k$: $$k^2(k^2+1)(k^2-1)=(k-1)k^2(k+1)(k^2+1)=5m $$ and consider it for $n=k+1$. We have $$(k+1)^2((k+1)^2+1)((k+1)^2-1)=\\ (k+1)^2(k^2+2k+2)(k^2+2k)=\\ k(k+1)^2(k+2)(k^2+2k+2).$$ Suppose this isn't a multiple of $5$. Then n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 10,
"answer_id": 4
} |
Help with solving mathematical induction problem I need help with the following:
Use mathematical induction to prove that for every $n\in N$,
$$
\sum_{k=1}^n\frac{1}{\cos kx \cos(k+1)x}=\frac{\tan(n+1)x-\tan x}{\sin x}
$$
For $n=1$, the statement is true.
Suppose that the statement is true for $n=m\in N$, and prove th... | $$\frac{\tan(m+2)x-\tan x}{\sin x}-\frac{\tan(m+1)x-\tan x}{\sin x}$$
$$=\frac{\tan(m+2)x-\tan(m+1)x}{\sin x}$$
$$=\dfrac{\sin[(m+2)x-\sin(m+1)x]}{\cos(m+2)x\cdot\cos(m+1)x\cdot \sin x}$$
$$=\dfrac1{\cos(m+2)x\cdot\cos(m+1)x}$$ if $\sin x\ne0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1262046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
need some help with a power series convergence test problem Find the interval of convergence for the given power series:
$$\sum\limits_{n=1}^\infty \frac{(x - 1)^n }{n(-4)^n}$$
First I applied the generalized ratio test, came out with $\frac{(1-x)}{4}$
Solved the inequality $|1-x| \lt 4$ and got $-3 \lt x \lt 5$.
But ... | $$\sum\limits_{n=1}^{\infty} \frac{(x - 1)^n }{n(-4)^n}$$
Using the ratio test, we have
$$ \lim\limits_{n\to\infty} \left|\frac{\frac{(x - 1)^{n+1}}{(n+1)(-4)^{n+1}}}{\frac{(x - 1)^n }{n(-4)^n}}\right|$$
$$ =\lim\limits_{n\to\infty} \left|\frac{(x - 1)^{n+1}n(-4)^n}{(x-1)^n(n+1)(-4)^{n+1}}\right|$$
$$ =\lim\limits_{n\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1264123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
A question on the Lagrange Inversion Formula I have to use the L.I.F. for
\begin{align*}
s\left(x,y\right)=\frac{1}{2}\left(1-x-y-\sqrt{1-2x-2y-2xy+x^2+y^2}\right)
\end{align*}
to obtain that
\begin{align*}
s\left(x,y\right) = \sum_{p,q\geq1}\frac{1}{p+q-1}\binom{p+q-1}{p}\binom{p+q-1}{q}x^py^q
\end{align*}
I've trie... | We seek to use Lagrange Inversion to show that
$$s(x,y) = \frac{1}{2}
\left(1-x-y-\sqrt{1-2x-2y-2xy+x^2+y^2}\right)$$
has the series expansion
$$\sum_{p,q\ge 1} \frac{1}{p+q-1}
{p+q-1\choose p} {p+q-1\choose q} x^p y^q.$$
On squaring we obtain
$$4 s(x,y)^2 = (1-x-y)^2 + 1-2x-2y-2xy+x^2+y^2
\\ - 2(1-x-y) (1-x-y-2s(x,y))... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1266250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$.
Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$.
Attempt:
$$ \int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy
= \int_0^1 \int_\sqrt{y}^1 x^2 + y^2 dx dy
= 1/3 + 1/3 - 2/15 - 2/7 = \frac{26}{105}.$$
However, the solution should be $26/35$ ... | There is nothing wrong with your calculations.
$$\int_0^1 \int_{y^{1/2}}^1 x^2 + y^2 \ dx \ dy = \int_0^1 y^2 - y^{5/2} + \frac13(1-y^{3/2}) \ dy \\
= \frac 13 - \frac 27 + \frac 13 - \frac 2{15} = \frac{26}{105}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find solutions to $\cot(x)+\csc(x)=\sqrt3$ in range $[0,2\pi]$ What is the best way to do the above? Are there any tricks I should be aware of.
I know how to simplify it to $\dfrac{\cos(x)}{\sin(x)} + \dfrac{1}{\sin(x)} = \sqrt{3}$
so multiplying both sides by $\sin(x)$, we get $\cos(x)+1=\sqrt{3}\sin(x)$.
But I'm stuc... | Let $x = 2t$, then we can start at where you left off:
$\cos x + 1 = \sqrt{3}\sin x\Rightarrow \cos (2t)+1 = \sqrt{3}\sin (2t) \Rightarrow 2\cos^2 t = 2\sqrt{3}\sin t\cos t \Rightarrow 2\cos t(\cos t - \sqrt{3}\sin t) = 0\Rightarrow \cos t = 0 \Rightarrow t = (2n+1)\dfrac{\pi}{2} \Rightarrow x = 2t = (2n+1)\pi, n \in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1269194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
Continued product in $\sin$ series Find the value of the product $$(\sin 1°)(\sin 3°)(\sin 5°)\ldots(\sin 89°)$$ I tried multiplying and dividing by $2$ and then combining and then converting into cosine, but doesn't work out.
| Notice for any integer $N > 0$, we have
$$\begin{align}
z^{2N} + 1
&= \prod_{k=-N}^{N-1} \left( z - e^{\frac{2k+1}{2N}\pi i} \right)
= \prod_{k=0}^{N-1}\left(z - e^{\frac{2k+1}{2N}\pi i}\right)\left(z - e^{-\frac{2k+1}{2N}\pi i}\right)\\
&= \prod_{k=0}^{N-1}\left[ z^2+1 - 2z\cos\left(\frac{2k+1}{2N}\pi\right)\right]
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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$f(x)$ is a polynomial satisfying $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2)$), given $f(2)=5.$
If f(x) is a polynomial satisfying $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2))$, given $f(2)=5.$
ATTEMPT:- $f(f(2))=f(5)$, We can find $f(0)$,$f(1)$ and $f(1/2)$ to be $1,2$ and $5/4$ respectively.
we can change the f... | Notice that you can get values for $f(2n) = (2n)^2 + 1$. Observing this, I suppose that $f$ is a quadratic polynomial with form $f(t) = at^2 + bt + 1$ ($+1$ since we know that $f(0) = 1$).
Writing the functional equation as $(f(x) - 1)(f(y) - 1) = f(xy) - 1$, we get $(ax^2 + bx)(ay^2 + by) = (ax^2 y^2 + bxy)$, so compa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1270658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to interpret relation of variables as eigenvector? I am trying to calculate the eigenvectors of a square matrix $A \in \mathbb{R}^{4x4}$.
$$A =
\begin{pmatrix}
a & 1 & 0 & 0 \\
0 & a & 1 & 0 \\
0 & 0 & a & 0 \\
0 & 0 & 0 & b
\end{pmatrix}
$$
To receive the eigenvalues I did:
$$\chi_A = \begin{vmatrix}
a-\lambda & ... | There are two cases to separate:
*
*If $a = b$, then you 'only' get one eigenvalue $a$ with multiplicity 4. In this case
$$\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & b-a
\end{bmatrix} = \begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Let $a,b$ be relative integers such that $2a+3b$ is divisible by $11$. Prove that $a^2-5b^2$ is also divisible by $11$. The divisibility for $11$ of $a^2 - 5b^2$ can be easily verified; in fact: $$a \equiv \frac {-3}{2}b \pmod {11}$$ therefore $$\frac {9}{4}\cdot b^2 - 5b^2 = 11(-\frac{b^2}{4}) \equiv 0 \pmod {11}.$$
T... | We have for some integer $c$ $$2a+3b=11c,$$ and this is equivalent to $$a=\frac{11c-3b}{2}.$$ Now, squaring both sides we get $$a^2=\frac{11^2c^2-66bc+9b^2}{4} $$ and substracting $5b^2$: $$a^2-5b^2=\frac{11^2c^2-66bc-11b^2}{4} $$ $$ a^2-5b^2=11\left(\frac{11c^2-6bc-b^2}{4}\right).\tag{$\star$}$$ Note that the quantity... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1271972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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$\sum \limits_{n \geq 0}a_n \frac{x^n}{n!}=e^{x+x^2/2}$ implies $a_n \sim \frac1{\sqrt2} n^{\frac n2}e^{ -\frac n2+\sqrt n -\frac14 }$
Prove the following asymptotic formula for the exponential generating function coefficients of $e^{x+x^2/2}$: $\; \; a_n \sim \frac1{\sqrt2} n^{\frac n2}e^{ -\frac n2+\sqrt n -\frac14 ... | Note: This answer follows section 5.4, example 2 of H.Wilf's Generatingfunctionology
Observe that $f(x)=e^{x+\frac{x^2}{2}}$ is an entire function which can therefore be written as power series
\begin{align*}
f(x)=\sum_{n\geq 0}a_nx^n
\end{align*}
converging everywhere in the complex plane. For this type of functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1272904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Probability Roulette Problem Suppose the Roulette table has 37 numbers (European Roulette table). During 37 spins, I always do the same bet: 35 numbers straight (35 chips in 35 different numbers).
Then:
*
*the probability of winning the 37 consecutive spins is $(\frac{35}{37})^{37}\approx 0.1279$,
*the probability... | Second question:
It has to be $Bin(n,\frac{2}{37}|1) > Bin(n,\frac{2}{37}|2)$.
${n \choose 1} \left( \frac{2}{37} \right)^1\cdot \left( \frac{35}{37} \right)^{n-1} >{n \choose 2} \left( \frac{2}{37} \right)^2\cdot \left( \frac{35}{37} \right)^{n-2}$
$n\cdot \frac{2}{37} \cdot \frac{37}{35} \cdot \left( \frac{35}{37} \r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1273353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$ for a 4-th degree monic polynomial If $f(x)=x^4+ax^3+bx^2+cx+d$.
Given $f(1)=10,f(2)=20,f(3)=30$ find $f(12)+f(-8)$.
This problem has troubled me a lot.The more I try to solve it,it becomes lengthier.
My problem is that there are four unknowns and only three equations.... | The fact that there are three equiations for four unknowns states that the solution will have one unconstrained unknown. Since you're asked for a concrete answer, the $f(12) + f(-8)$ is a specially crafted expression that doesn't depend that particular unconstrained unknown.
Let's simplifiy things by introducing $y = x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1274133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.