Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
$x^a - 1$ divides $x^b - 1$ if and only if $a$ divides $b$ Let $x > 1$ and $a$, $b$ be positive integers. I know that $a$ divides $b$ implies that $x^a - 1$ divides $x^b - 1$. If $b = qa$, then
$$x^b - 1 = (x^a)^q - 1^q = (x^a - 1)((x^a)^{q-1} + \ldots + x^a + 1).$$
I'm interested in the converse of the statement. If $... | Let $b=a \cdot q+r$, where $0 < r < a$.
Then
$$x^b-1=x^b-x^r+x^r-1=x^r(x^{aq}-1)+x^r-1 \,.$$
Use that $x^a-1$ divides $x^{aq}-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/128007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 2,
"answer_id": 1
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Matrix multiplication proof: $(A+B)^k$ for matrices such that $A^2=B^2=0$ and $AB=BA$ I have the following details about the matrices $A,B$:
$$A_{n\times n},B_{n\times n}$$ $$A^2 = B^2 = 0$$ $$AB=BA$$
I need to find $x \in \mathbb{N}$ so $(A+B)^x=0$
What implications can I make from the given details?
*
*$A,B$ doe... | $$\mbox{ let } x > 2\mbox{ and } x\in \mathbb{N}$$
$$\textbf{(A+B)}^2 = \textbf{A}^2+\textbf{B}^2+2\textbf{AB} = \textbf{2AB}$$
$$\textbf{(A+B)}^x = \textbf{(A+B)}^2\textbf{(A+B)}^{x-2}$$
$$\Rightarrow \textbf{(A+B)}^x =2\textbf{AB}\textbf{(A+B)}^{x-2}$$
$$\Rightarrow \textbf{(A+B)}^x =2\textbf{AB}\textbf{(A+B)}\textb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $n^5 - n$ is divisible by $30$ without reduction How can I prove that prove $n^5 - n$ is divisible by $30$?
I took $n^5 - n$ and got $n(n-1)(n+1)(n^2+1)$
Now, $n(n-1)(n+1)$ is divisible by $6$.
Next I need to show that $n(n-1)(n+1)(n^2+1)$ is divisible by $5$.
My guess is using Fermat's little theorem but ... | Funny brute force approach. Just show:
$$n^5-n = 5!\binom{n+2}{5} + 5\cdot 3!\binom{n+1}{3}=120\binom{n+2}{5}+30\binom{n+1}{3}$$
In general, if $d$ is odd, then you can write:
$$n^d-n = \sum_{i=0}^{\lfloor d/2\rfloor} a_i \binom{n+i}{2i+1}$$
where the $a_i$ are multiples of $(2i+1)!$.
Since the left side is divisible b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/132210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 21,
"answer_id": 8
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Solving $\lim\limits_{n\to\infty} \frac{\cot{\frac{2}{n}}+n\csc{\frac{3}{n^3}}}{\csc{\frac{3}{n}} + n\cot{\frac{2}{n^2}}}$ How to get from
$$\lim_{n\to\infty} \frac{\cot{\frac{2}{n}}+n\csc{\frac{3}{n^3}}}{\csc{\frac{3}{n}} + n\cot{\frac{2}{n^2}}} = \lim_{n\to\infty} \frac{\frac{\frac{2}{n}}{\tan{\frac{2}{n}}}\cdot\fra... | I think you have to check your question because i don't think answer is $\frac{2}{3}$. But i am given a solution of that question which you are write here.
We know that $\displaystyle \lim_{x\to 0}\frac{x}{\sin x}=1$ and $\displaystyle \lim_{x\to 0}\frac{x}{\tan x}=1$.
$\displaystyle \lim_{n\to\infty}\frac{\cot \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the linear combinations of two vectors I am studying for my finals and I'm trying to answer the following question:
Consider the following two vectors in $\mathbb{R}^3$: $a=(1,2,3)$ and $b=(2,3,1)$. Decide
whether it is possible to express the vector $c=(2,4,5)$ as a linear
combination of $a$ and $b$.
I ... | You can also do your example in following way,
Let $(2,4,5)=\alpha.(1,2,3)+\beta.(2,3,1)$
$(2,4,5)=(\alpha+2\beta,2\alpha+3\beta,3\alpha+\beta)$
$\therefore 2=\alpha+2\beta,4=2\alpha+3\beta,5=3\alpha+\beta$
Now, solving the first two equation, we get, $\alpha=2,\beta=0$ but it does not satisfied last equation therefore... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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Trapezoid area and wrong answer Knowing that the acute angles of the trapezoid are $60^\circ$ and $45^\circ$ and the difference of the squares of base lenghts is equal to 100, calculate the area of this trapezoid.
Here's my solution: Let a be the shorter and b the longer base. By drawing two lines from the ends of a pe... | I noted things in the following order:
*
*$b^2-a^2=(b-a)(b+a)=100$ so $A=\frac{h(b+a)}{2}=\frac{50h}{b-a}$.
*Then I reasoned that $b-a=h+\frac{h}{\sqrt{3}}$.
*Combining them yields $A=\frac{50h}{h+\frac{h}{\sqrt{3}}}=\frac{50\sqrt{3}}{\sqrt{3}+1}=\frac{50\sqrt{3}(\sqrt{3}-1)}{2}=25(3-\sqrt{3})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/140479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Derivative of $\ln(x\sqrt{x^2-1})$ I am trying to find the derivative of $\ln(x\sqrt{x^2-1})$ but I can not get what the book gets.
I get $$\frac{1}{x \sqrt{x^2-1}} \cdot \sqrt{x^2-1} + x\cdot\frac{1}{2}(x^2-1)^\frac{-1}{2}\cdot2x$$
which I reduce to
$$\begin{align}
&\frac{1}{x\sqrt{x^2-1}}\sqrt{x^2-1} + x^2(x^2-1)^\fr... | You forgot a bracket:
$$\frac{1}{x \sqrt{x^2-1}} * \left[ \sqrt{x^2-1} + x*\frac{1}{2}(x^2-1)^\frac{-1}{2}2x \right]$$
Also, might be much easier to use properties of Log:
$$\ln(x\sqrt{x^2-1}) = \ln(x) +\frac{1}{2} \ln(x^2-1) \,$$
This is much easier to differentiate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/143550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Evaluating $\int_0^a \frac{x^{b}+x^{c}}{(1+x)^{b+c+2}}$ Let $0 <a < 1$ and let $b,c \in \mathbb{N}$, evaluate $$\int_0^a \frac{x^{b}+x^{c}}{(1+x)^{b+c+2}}$$
How to evaluate in terms of $a$,$b$ and $c$?
| Write the integral as
$$I(a,b,c)=\int_0^a {{{\left( {\frac{x}{{1 + x}}} \right)}^b}{{\left( {\frac{1}{{1 + x}}} \right)}^c}\frac{{dx}}{{{{\left( {1 + x} \right)}^2}}}} + \int_0^a {{{\left( {\frac{x}{{1 + x}}} \right)}^c}{{\left( {\frac{1}{{1 + x}}} \right)}^b}\frac{{dx}}{{{{\left( {1 + x} \right)}^2}}}} $$
Since $$\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/144982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluating $ \int\limits_{0}^{2 \pi} \cos( n \arctan( R_0 \sin(\vartheta))) \cos( n \arctan( R_0 \cos(\vartheta))) \; d \vartheta.$ Let $R_0$ be a real number and $n$ an arbitrary integer
$$ \int\limits_{0}^{2 \pi} \cos( n \arctan( R_0 \sin(\vartheta))) \cos( n \arctan( R_0 \cos(\vartheta))) \; d \vartheta.$$
I have tr... | Note that, using invariance of the integrand under $\theta \to \frac{\pi}{2} + \theta$ for $0 < \theta < \frac{\pi}{2}$:
$$
\mathcal{I}_n(\rho)= \int_0^{2 \pi} \cos\left( n \arctan(\rho \sin(\theta) ) \right) \cos\left( n \arctan(\rho \cos(\theta) ) \right) \mathrm{d} \theta = \\4 \int_0^{\pi/2} \cos\left( n \arctan(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/145603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Finding the integral of $(1+2^{2x})/2^x$
Evaluate the integral:
$$\int\frac{1+2^{2x}}{2^x}\,dx = \int \frac{ 1 + (2^x)^2}{2^x}\,dx$$
Let $u = 2^x$. Then $du = 2^x\ln2\,dx$, which yields $\frac{du}{2^x\ln2} = dx$ so
$$ \int \frac{ 1 + (2^x)^2}{2^x}\,dx = \int \frac{1+u^2}{u}du =
\left( x+ \frac{u^3}{3} \right)\ln... | All you need is that $$\int a^x dx = \int \exp(x \log a) dx = \frac{\exp(x \log a)}{\log a} +C= \frac{a^x}{\log(a)}+C$$
$$\begin{align*}
\int \frac{1+2^{2x}}{2^{x}} dx &= \int \left(\left(\frac12 \right)^x + 2^x \right)dx\\
&= \left( \frac{\left(\frac12 \right)^x}{\log(1/2)} + \frac{2^x}{\log(2)} \right) + C\\
&= \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If an integer number is a square and a cube, then it can be writen as $5n,5n+1$, or $5n+4$ I want to show that if an integer number is a square and a cube, then it can be writen as $5n,5n+1$, or $5n+4$.
I tried the following. There are integers numbers $x,y$ such that $n=x^{2}=y^{3}.$ By using Euclidean division, then... | Already if an integer is a square it must be of the shapes you have described.
It is easiest to prove this using congruence notation. Any integer is congruent to $0$, $1$, $2$, $3$, or $4$ modulo $5$. Note now that $0^2$, $1^2$, $2^2$, $3^2$, and $4^2$ are respectively congruent to $0$, $1$, $4$, $4$, and $1$ modulo $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
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Linear transform of polynomials The following question is from Golan's linear algebra book. I have posted a solution in the answers.
Problem: Let $F$ ba field and let $V$ be a vector subspace of $F[x]$ consisting of all polynomials of degree at most 2. Let $\alpha:V\rightarrow F[x]$ be a linear transformation satisfyi... | The idea here is to determine the action on each of the basis elements $1$, $x$, and $x^2$.. By linearity we see
$\alpha(x)=\alpha((x+1)-1)=\alpha(x+1)-\alpha(1)=x^5+x^3-x$
$\alpha(x^2)=\alpha((x^2+x+1)-(x+1))=\alpha(x^2+x+1)-\alpha(x+1)=-x^5+x^4-x^3-x^2+1$
Using linearity again we see
$\alpha(x^2-x)=\alpha(x^2)-\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How can we produce another geek clock with a different pair of numbers? So I found this geek clock and I think that it's pretty cool.
I'm just wondering if it is possible to achieve the same but with another number.
So here is the problem:
We want to find a number $n \in \mathbb{Z}$ that will be used exactly $k \in \... | For $n=2$ and $k=12$ here is a solution:
$1=\left(2 \times \left(2 \times \left(2 \times \frac{2}{\left(2+\left(2+\left(2+\left(2+\left(2+\left(2+\left(2+2\right)\right)\right)\right)\right)\right)\right)}\right)\right)\right)$
$2=\left(2+\left(2 \times \left(2+\left(2+\left(2+\left(2+\left(2-\left(2+\left(2+\left(2+\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 10,
"answer_id": 4
} |
simplify $ (-2 + 2\sqrt3i)^{\frac{3}{2}} $? How can I simplify $ (-2 + 2\sqrt3i)^{\frac{3}{2}} $ to rectangular form $z = a+bi$?
(Note: Wolfram Alpha says the answer is $z=-8$. My professor says the answer is $z=\pm8$.)
I've tried to figure this out for a couple hours now, but I'm getting nowhere.
Any help is much app... | $\:\sqrt{-2+2\sqrt{-3}}\:$ can be denested by a radical denesting formula that I discovered as a teenager.
Simple Denesting Rule $\rm\ \ \ \color{blue}{subtract\ out}\ \sqrt{norm}\:,\ \ then\ \ \color{brown}{divide\ out}\ \sqrt{trace} $
Recall $\rm\: w = a + b\sqrt{n}\: $ has norm $\rm =\: w\:\cdot\: w' = (a + b\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/152989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I compute mean curvature in cylindrical coordinates? If I have a surface defined by $ z=f(r, \theta) $, does anyone know the expression for the mean curvature? There is a previous post dealing with Gaussian instead of mean curvature, the answer I'm looking for is similar to that given by J.M. on that post.
The m... | It's similar to the Gaussian Curvature. J.M. actually hide his/her calculation.
You start with the parametrization:
\begin{align}
x &= \rho(\vartheta, z)\cos\vartheta \\
y &= \rho(\vartheta, z)\sin\vartheta \\
z &= z \\
\end{align}
you need to find the values of $L$, $M$ and $N$ of the Second fundamental form and then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Integral of $\int \frac {\sqrt {x^2 - 4}}{x} dx$ I am trying to find $$\int \frac {\sqrt {x^2 - 4}}{x} dx$$
I make $x = 2 \sec\theta$
$$\int \frac {\sqrt {4(\sec^2 \theta - 1)}}{x} dx$$
$$\int \frac {\sqrt {4\tan^2 \theta}}{x} dx$$
$$\int \frac {2\tan \theta}{x} dx$$
From here I am not too sure what to do but I know I ... | You are correct. First note that you have not carried a factor of $2$, since your integral is $2 \int \tan^2(\theta) d \theta$.
Hence your solution should read $$2 \tan(\text{arsec}(x/2)) - 2 \text{arcsec}(x/2) + c$$
You may want to rewrite your solution to match with the solution in your text. For instance, $$\text{ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Simplify this expression with nested radical signs My question is-
Simplify:
$$\frac1{\sqrt{12-2\sqrt{35}}}-\frac2{\sqrt{10+2\sqrt{21}}}-\frac1{\sqrt{8+2\sqrt{15}}}$$
| $$
\begin{align} & {}\quad \frac1{\sqrt{12-2\sqrt{35}}}-\frac2{\sqrt{10+2\sqrt{21}}}-\frac1{\sqrt{8+2\sqrt{15}}}\\[10pt]
& =\frac {1}{\sqrt{ 12-2 \sqrt {35}}} \frac {\sqrt{ 12+2 \sqrt {35}}}{\sqrt{ 12+2 \sqrt {35}}}- \frac {2 }{\sqrt{ 10+2 \sqrt {21}}} \frac {\sqrt{ 10-2 \sqrt {21}}}{\sqrt{ 10-2 \sqrt {21}}}- \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Putting ${n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ in a closed form As the title says, I'm trying to transform $\displaystyle{n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ into a closed form. My work:
$\displaystyle\left(1 + \exp\frac{2i\pi}{5... | Hint: with $\omega=\exp(2\pi i /5)$ a primitive $5$th root of unity, we have
$$\sum_{r=0}^4 \omega^{rk}=\begin{cases}5 & k\equiv 0 \bmod 5 \\ 0 & k\not\equiv 0\bmod 5.\end{cases}$$
So then what is
$$(1+\omega^0)^n+(1+\omega^1)^n+(1+\omega^2)^n+(1+\omega^3)^n+(1+\omega^4)^n~? $$
(Combine the binomial expansions...)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
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Is this string ascending or descending? I need help in checking that string is ascending or descending and I have problems with it.
so it's my "$a_n$"
$$
\sqrt {n+5} - \sqrt{n}
$$
and I need to use $a_{n+1} - a_n$ or $\frac {a_n+1}{a_n}$.
EDIT:
it would be something like that
$$
\sqrt {n+6} - \sqrt{n+1} -\sqrt{n+5}+\... | \begin{align*}
a_n &= \sqrt{n+5}-\sqrt{n} = \left(\sqrt{n+5}-\sqrt{n}\right)\frac{\sqrt{n+5}+\sqrt{n}}{\sqrt{n+5}+\sqrt{n}} \\
&= \frac{\left(\sqrt{n+5}-\sqrt{n}\right)\left(\sqrt{n+5}+\sqrt{n}\right)}{\sqrt{n+5}+\sqrt{n}} \\
&= \frac{n+5 - n}{\sqrt{n+5}+\sqrt{n}} \\
&= \frac{5}{\sqrt{n+5}+\sqrt{n}} \\
\end{align*}
Thi... | {
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"timestamp": "2023-03-29T00:00:00",
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What other substitutions could I use to evaluate this integral? Consider the integral
$$ \int x^2\sqrt{2 + x} \, dx$$
I need to find the value of this integral, yet all its (seemingly) possible substitutions don't allow me to cancel appropriate terms. Here are three substitutions and their outcomes, all of which cover ... | If $u=\sqrt{2+x}$ then $u^2 = 2+x$, so $2u\,du=dx$, and $x=u^2-2$. Then you have
$$
\int x^2\sqrt{2+x}\,dx = \int(u^2-2)^2 u\, 2u\,du
$$
After expanding the polynomial and getting its antiderivative, put $\sqrt{2+x}$ in place of $u$ wherever it appears.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/157548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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How to solve this quadratic congruence equation How to solve $x^2+x+1\equiv 0\pmod {11}$ ?
I know that in some equations like $ax\equiv b\pmod d$ if $(a,d)=1$ then the equation has one and only one solution $x\equiv ba^{\phi(d)-1}\pmod d$. Any help will be appreciated. ;)
| You can use the technique of Completing the Square.
Suppose that we want to solve the congruence $ax^2+bx+c\equiv 0 \pmod p$, where $a\not\equiv 0\pmod{p}$, and $p$ is an odd prime. The congruence is equivalent to
$$4a^2x^2+4abx+4ac\equiv 0\pmod{p}.\tag{$1$}$$
Completing the square, we see that the congruence is equiva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 0
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the probability of picking certain amount of items from a sack. I need a help on calculating the probability. Here is the task that I can "translate" it into a probability task.
Let's assume we have a sack with 12 items. There are 1 X item and 2 Y items and other staff. Every time we pick an item from the sack and thro... | Ok then. Your sack of $12$ items has $1$ item of type 'X', $2$ items of type 'Y', and $9$ other items. Suppose you draw an item from the sack $n$ times.
To not have at least $2$ X and at least $3$ Y, you must have picked either:
*
*$0$ X and any number of $Y$ (ways: $11^n$),
*$1$ X and any number of $Y$ (ways: ${n ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two algebra questions I have two questions that I need help with:
1) How many single digit even natural number solutions are there for the equation $A+B+C+D = 24$ such that $A+B > C+D$
A)20 B)11 C)16 D)24
2) Three positive real numbers $x,y,z$ are such that $x+y+Z = 1$. which of the following inequalities best discribe... | For the inequality note that:
$$(x-y)^2 + (y-z)^2+(z-x)^2 = 2(x^2+y^2+z^2)-2(xy+yz+zx) \geq 0$$
so that $$x^2+y^2+z^2 \geq xy+yz+zx$$
Now
$$1=(x+y+z)^2=(x^2+y^2+z^2) + 2(xy+yz+zx)\geq 3(xy+yz+zx)$$
I think that is pre-calculus algebra, and not nearly as well known a trick as it should be.
| {
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"url": "https://math.stackexchange.com/questions/162949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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How to find $ \lim\limits_{ x\to 100 } \frac { 10-\sqrt { x } }{ x-100 }$ Find $ \lim\limits_{ x\to 100 } \dfrac { 10-\sqrt { x } }{ x-100 }$
(without using a calculator and other machines...?)
| Recognizing that $\sqrt{x} \approx 10$ makes the numerator vanish, we may be inspired to use a differential approximation:
$$ \sqrt{x} = 10 + \frac{1}{20}(x - 100) + r(x) (x - 100) $$
where $r(x)$ has the property that $\lim_{x \to 100} r(x) = 0$. Therefore,
$$ \lim_{x \to 100} \frac{10 - \sqrt{x}}{x - 100}
= \lim_{x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 6
} |
$X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$ $X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$
I changed the bases to 10, then performed manual addition/multiplication but it didn't yield me any result except for long terms. Please show me the way.
All I'm getting is $$\frac{\lg 18\lg54 +... | $ X = \log_{12} 18 $ and $ Y= \log_{24} 54 $.
$ X = \frac {\log_{2}18}{\log_{2}12}=\frac{\log_{2}9\cdot2}{\log_{2}4\cdot3}=\frac{\log_{2}3^2\cdot 2}{\log_{2}2^2\cdot3}=\frac{2\log_{2}3 + 1}{\log_{2}3 + 2}=\frac{2A+1}{A+2}$
$ Y = \frac {\log_{2}54}{\log_{2}18}=\frac{\log_{2}27\cdot2}{\log_{2}8\cdot3}=\frac{\log_{2}3^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
All integer solutions for $x^4-y^4=15$ I'm trying to find all the integer solutions for $x^4-y^4=15$.
I know that the options are $x^2-y^2=5, x^2+y^2=3$, or $x^2-y^2=1, x^2+y^2=15$, or $x^2-y^2=15, x^2+y^2=1$, and the last one $x^2-y^2=3, x^2+y^2=5$.
Only the last one is valid. $x^2+y^2=15$ is not solvable since the p... | Consider $f(x) = (x+1)^4 - x^4 = 4x^3 + 6x^2 + 4x + 1$. Then $f'(x) = 12x^2 + 12x + 4 = 3(2x+1)^2 + 1$, which is always positive. So f(x) is always increasing, meaning that for y more than 1 we don't have any valid solutions. Similarly you can go into the negatives and see that y cannot be less than -1. Now you only ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/166070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
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Recurrence relation $C_n = n + 1 + \dfrac{2}{n}\sum\limits_{k=0}^{n-1}C_k$. A Discrete Mathematics book from which I'm self-studying ("Discrete Mathematics and Its Applications", by Kenneth Rosen) asks me to do the following:
Given the following recurrence relation:
$$C_n = n + 1 + \frac{2}{n}\sum_{k=0}^{n-1}C_k$$
The ... | $C_n=n+1+(2/n)\sum^{n-1}$; $nC_n=n^2+n+2\sum^{n-1}$; $$nC_n-(n+1)C_{n-1}=n^2+n-((n-1)C_n-2\sum^{n-2})=n^2+n-((n-1)^2+(n-1))=2n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/168214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Fastest way to compare fractions Which is the fastest method to compare the below fractions with minimum calculation possible and finding which is greatest and which the smallest??
$$\frac{26}{686},\quad \frac{48}{874},\quad \frac{80}{892},\quad \frac{27}{865}$$
| $\dfrac{1}{30} < \dfrac{26}{686} < \dfrac{1}{20}$ because $30 \times 26 = 780 > 686$ while $20 \times 26 = 520 < 686$.
$\dfrac{1}{20} < \dfrac{48}{874} < \dfrac{1}{12}$ because $20 \times 48 = 960 > 874$ while
$12 \times 48 = 144 \times 4 = 576 < 874$
$\dfrac{1}{12} < \dfrac{80}{892}$ because $12 \times 80 = 960 > 892$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$? Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?
| Recall either of Euler's two famous expressions for $\sin x$:
$$\sin x=x\prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right),$$
or
$$\sin x=x\prod_{n=1}^\infty\left(1-\frac{x^2}{\pi^2n^2}\right).$$
Now let $x=\dfrac{\pi}{6}$.
Or else use the following formula of Viète
$$\frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 7,
"answer_id": 2
} |
Showing that the last digit of $a$ and $a^{13}$ are the same
For $a \in \mathbb N$, show that the last digit of $a$ and $a^{13}$
are the same.
For example: $2^{13} = 8,192$
$7^{13} = 96,889,010,407$
| Claim: For all integers $a$ we have $10|a^5-a$.
This follows from Fermat Little Theorem, but can also be proven elementary.
Proof of the claim
$$a^5-a=a(a-1)(a+1)(a^2+1) \,.$$
Now, one of $a$ or $a+1$ is even thus $a^5-a$ is divisible by $2$.
Also
$$a^5-a=a(a-1)(a+1)(a^2-4+5)=a(a-1)(a+1)(a^2-4)+5a(a-1)(a+1)$$
$$a^5-a=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 6
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Finding a number when its remainder is given. I am trying to solve this problem
W is a positive integer when divided by 5 gives remainder 1 and when divided by 7 gives remainder 5. Find W.
I am referring back to an earlier post I made. Now I am attempting to solve it that way.
We know that
$$w\equiv1(mod~5)$$
$$w\... | w is of the form 5a+1=7b+5 where a,b are integers.
=>5a=7b+4
=>5a+10=7b+14
=>5(a+10)=7(b+2)
=>5 divides (b+2) as (5,7)=1
=>b is of the form 5c-2 where c is any integer.
w=7b+5=7(5c-2)+5=35c-9=35d+26 where d=c-1
Alternatively, according to Euclid's GCD algorithm,
there exists integers c,d such that cx+dy=(x,y).
As (5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Evaluate :$\int \frac{\sin^{-1} \sqrt{x} -\cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}} dx$ How to evaluate
$$
\int \frac{\sin^{-1} \sqrt{x} -\cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}} dx
$$
I know that $\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}=\frac{\pi}{2}$ but after that I have no idea, ... | *
*Using the relation $\arcsin(\sqrt{x}) + \arccos(\sqrt{x}) = \frac{\pi}{2}$ (which is valid for $0 \leqslant x \leqslant 1$, so this must an implicit assumption in your problem) solve for $\arccos(\sqrt{x})$ and substitute that into the integrand.
*After that make a $u$-substitution $u = \arcsin(\sqrt{x})$. This s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
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Looking for multiple solutions to show my students. Let ABC be a right triangle with B the right angle. X,Y and Z are on BC, CA and AB respectively such that BXYZ is a square. If the square is of side length m, AY = r and YC = s, find m in terms of r and s.
I have two solutions for my students (I instruct a math team c... | The solution for someone who really prefers algebra to geometry:
Let $AZ=p$, $XC=q$. Then the Pythagorean theorem tells us that
\begin{eqnarray}
p^2+m^2&=&r^2\\
q^2+m^2&=&s^2\\
(p+m)^2+(q+m)^2&=&(r+s)^2 \, .
\end{eqnarray}
Subtracting the first two equations from the third and simplifying yields
$p+q=\frac{rs}{m}$, whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
| Given, $x^2f(x)+f(1−x)=2x−x^4$
$f(-1) = 0$
$f(0) = 1$
$f(1) = 0$
$f(2) = -3$
$f(3) = - 8$
$f(4) = -15$
By careful observation one can see that the difference between the terms vanishes after the second stage
i.e. $f(x)$ is of the form $a{x^2} + b{x} +c$
i.e. $f(x) = a{x^2} + b{x} + c$ -------(1)
i.e. $f(0) = c = 1$
i.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 2
} |
Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$ Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$.
Can anyone help me with this? Thank You!
| Here's a linear algebraic route: from this answer, we find that the eigenvalues of the $4\times4$ min-matrix
$$\mathbf M=\begin{pmatrix}
1 & 1 & 1 & 1 \\
1 & 2 & 2 & 2 \\
1 & 2 & 3 & 3 \\
1 & 2 & 3 & 4
\end{pmatrix}$$
are $\lambda_k=\dfrac14\sec^2\left(\dfrac{k\pi}{9}\right)$ for $k=1,\dots,4$. From this, we have t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 1
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Compute $ I_{n}=\int_{-\infty}^\infty \frac{1-\cos x \cos 2x \cdots \cos nx}{x^2}\,dx$ I'm very curious about the ways I may compute the following integral. I'd be very glad to know your approaching ways for this integral:
$$
I_{n} \equiv
\int_{-\infty}^\infty
{1-\cos\left(x\right)\cos\left(2x\right)\ldots\cos\left(nx\... | First note that
$$\int_{-\infty}^{\infty} \frac{1-\cos ax}{x^2} \; dx
= \left[ -\frac{1-\cos ax}{x}\right]_{-\infty}^{\infty} + a \int_{-\infty}^{\infty} \frac{\sin ax}{x} \; dx
= \pi \, |a|,$$
by the Dirichlet integral. Also, by mathematical induction we can easily prove that
$$ \prod_{k=1}^{n} \cos \theta_k = \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
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"answer_id": 0
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Prove $\sin^2(A)+\sin^2(B)-\sin^2(C)=2\sin(A)\sin(B) \cos(C)$ if $A+B+C=180$ degrees I most humbly beseech help for this question.
If $A+B+C=180$ degrees, then prove
$$
\sin^2(A)+\sin^2(B)-\sin^2(C)=2\sin(A)\sin(B) \cos(C)
$$
I am not sure what trig identity I should use to begin this problem.
| This may not be the shortest way, but is pretty systematic and doesn't involve any real trickery. First, we eliminate the $C$ angles from the equation, using that $C = 180 - A - B$. We write $\sin(C) = \sin(180 - A - B) = \sin(A + B)$, and $\cos(C) = \cos(180 - A - B) = -\cos(A + B)$ and what you need to prove is
$$\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 3
} |
a question about an infinite series calculation. I want to prove that for $y >0$, $ x \in \mathbb R$,
$$ \sum_{n=-\infty}^\infty \frac{y}{(x+n)^2 + y^2} = \frac{1}{2} \frac{1 - e^{-4 \pi y }}{1 - 2 e^{-2 \pi y} \cos ( 2 \pi x ) + e^{-4 \pi y}}$$
| Let's use J.M.'s excellent hint :
$$\sum_{n=-\infty}^\infty \frac{y}{(x+n)^2 + y^2}=-\ \Im{\sum_{n=-\infty}^\infty \frac 1{x+iy+n}}$$
Setting $z:=x+iy\ $ we will evaluate :
$$\sum_{n=-\infty}^\infty \frac 1{z+n}=\frac 1z+\sum_{n=1}^\infty \frac 1{z+n}+\frac 1{z-n}=\frac 1z+\sum_{n=1}^\infty \frac {2z}{z^2-n^2}$$
The se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Number of bit strings with 3 consecutive zeros or 4 consecutive 1s I am trying to count the number of bit-strings of length 8 with 3 consecutive zeros or 4 consecutive ones. I was able to calculate it, but I am overcounting. The correct answer is $147$, I got $148$.
I calculated it as follows:
Number of strings with 3 ... | Let $n$ be a nonnegative integer. Let $a_n$ be the number of bit strings of length $n$ with at least 3 consecutive zeros. Clearly $a_0 = 0$, $a_1 = 0$, $a_2 = 0.$ Let $n \geq 3.$ Denote by $S$ the set of all bit strings of length $n$ with at least 3 consecutive zeros. The set $S$ is a disjoint union of the following f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
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Proving $\sqrt{n}+\frac{1}{\sqrt{n+1}} \geq \sqrt{n+1}$ I would like to know how to prove the following assertion :
For every $n>0$: $$\sqrt{n}+\frac{1}{\sqrt{n+1}} \geq \sqrt{n+1}$$
| There are many approaches. One of them is to try to prove the equivalent assertion that $\sqrt{n+1}-\sqrt{n} \le \frac{1}{\sqrt{n+1}}$.
Note that
$$\sqrt{n+1}-\sqrt{n}=\frac{(\sqrt{n+1}-\sqrt{n})(\sqrt{n+1}+\sqrt{n})}{\sqrt{n+1}+\sqrt{n}}=\frac{1}{\sqrt{n+1}+\sqrt{n}}.\tag{$1$}$$
It is clear that for $n \gt 0$, the ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/179194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Evaluating $\int_0^1{\frac{1}{(x+3)^2}}\ln\left(\frac{x+1}{x+3}\right)dx$ using $\frac{dy}{dx}=\frac{2}{(x+3)^2}$ where $y=\frac{x+1}{x+3}$ Find derivative of $$y= \frac{ax+b}{cx+d}$$
I found it to be $$\frac{dy}{dx}=\frac{a}{cx+d}-\frac{c(ax+b)}{(cx+d)^2}$$
Use it to evaluate:
$$\int_0^1{\frac{1}{(x+3)^2}}\ln\left(\fr... | *
*You have:
$$\dfrac{dy}{dx}=\dfrac{(x+3)}{(x+3)^2}-\dfrac{(x+1)}{(x+3)^2}=\dfrac{2}{(x+3)^2}$$
But the integral is $I=\int_0^1{\dfrac{1}{(x+3)^2}}\ln\left(\dfrac{x+1}{x+3}\right)dx$ where ${\dfrac{1}{(x+3)^2}}$ is actually $\dfrac 12 \times \dfrac{2}{(x+3)^2}$. Therefore:
$$I= \dfrac 12 \int_0^1{\frac{dy}{dx}}\ln(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/179475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$? Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality
$$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?
| This is as far as I got...
$\frac{(1-b)\sqrt{a}}{(1-b)(1-a)}$ + $\frac{(1-a)\sqrt{b}}{(1-a)(1-b)}$ + $\frac{\sqrt{c}}{(1-c)}$ $\geq$ $\frac{3\sqrt{3}}{2} $
$\Leftrightarrow$
$\frac{(1-b)\sqrt{a} + (1-a)\sqrt{b}}{(1-b)(1-a)}$ + $\frac{\sqrt{c}}{(1-c)}$ $\geq$ $\frac{3\sqrt{3}}{2} $
$\Leftrightarrow$
$\frac{(1-c)(1-b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 2
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Compute $\lim\limits_{n\to\infty} \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{{1}/{2}}\cdots\left(1+\frac{n}{n}\right)^{{1}/{n}}$ Compute the limit:
$$\lim_{n\to\infty} \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{\frac{1}{2}}\cdots\left(1+\frac{n}{n}\right)^{\frac{1}{n}}$$
| We want to calculate $$\lim_{n \to \infty} \prod_{1 \leqslant k \leqslant n} \left(1 + \frac {k} {n}\right)^{\frac {1} {k}}.$$ Denote it by $\ell$. Taking logarithms we have
$$\begin {eqnarray}
\log \ell & = & \lim_{n \to \infty} \sum_{1 \leqslant k \leqslant n} \frac {1} {k} \log \left(1 + \frac {k} {n}\right) \nonumb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 2,
"answer_id": 1
} |
is there a method to find all the solutions to, $y^2-6y+\sqrt{y}+4=0$ is there a method to find all the solutions to the following set of irrational equations,
$\sqrt{x}+y=3$
$x+\sqrt{y}=5$
NOTE: $(4-1)=(2-1)(2+1)=3$ and $(4+1)=(2^2+1^2)=(3^2-2^2)=(3-2)(3+2)=5$
| $(3-y)^2+\sqrt{y}=5$
$y^2-6y+\sqrt{y}+4=0$
$(\sqrt{y}-1)(y\sqrt{y}+y-5\sqrt{y}-4)=0$
i.e. $\sqrt{y}=1$, and $x=4$ is a solution
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/185042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Properties of ellipse x-y form $$5x^2+8xy+5y^2=1$$
$$1\left(\frac{x-y}{\sqrt{2}}\right)^2+9\left(\frac{x+y}{\sqrt{2}}\right)^2=1$$
I know that these two forms are equal, showing that the equation is an ellipse.
I do know what happens when the ellipse is in the form of $ax^2 + by^2 = 1$ but not sure what happens in the... | Using this, if we rotate the axes $\theta $ as the rotation of axes does not change the nature of the curve,
$x=X\cos\theta-Y\sin\theta$, $y=X\sin\theta+Y\cos\theta$,
$5(X\cos\theta-Y\sin\theta)^2+5(X\sin\theta+Y\cos\theta)^2+8(X\cos\theta-Y\sin\theta)(X\sin\theta+Y\cos\theta)=1$
$=>X^2(5+8\sin\theta\cos\theta)+Y^2(5-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Sum of series $\sin x + \sin 2x + \sin 3x + \cdots $ Please help me compute the sum of the series:
$$\sin(x)+\sin(2x)+\sin(3x)+\cdots$$
| $$2\sin\frac{x}{2}\sin rx=\cos\frac{(2r-1)}{2}x-\cos\frac{(2r+1)}{2}x$$
Putting $r=1,2,\ldots,n-1,n$ we get,
$$2\sin\frac{x}{2}\sin x=\cos\frac{1}{2}x-\cos\frac{3}{2}x$$
$$2\sin\frac{x}{2}\sin 2x=\cos\frac{3}{2}x-\cos\frac{5}{2}x$$
$$\vdots$$
$$2\sin\frac{x}{2}\sin rx=\cos\frac{(2n-3)}{2}x-\cos\frac{(2n-1)}{2}x$$
$$2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
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Writing 1/3 as a sum of other numbers Is it possible to write $0.3333(3)=\frac{1}{3}$ as sum of $\frac{1}{4} + \cdots + \cdots\frac{1}{512} + \cdots$ so that denominator is a power of $2$ and always different?
As far as I can prove, it should be an infinite series, but I can be wrong.
In case if it can't be written usi... | Archimedes showed that if you have a finite sum in which each term is $1/4$ of the previous term, except that the last term is $1/3$ of the previous term, then the sum does not depend on the number of terms, but is just $4/3$ of the first term. In modern terminology:
$$
1+\frac 1 4 + \frac{1}{16} + \cdots + \frac{1}{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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how can one solve for $x$, $x =\sqrt{2+\sqrt{2+\sqrt{2\cdots }}}$
Possible Duplicate:
Limit of the nested radical $\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$
how can one solve for $x$, $x =\sqrt[]{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2\cdots }}}}}}$
we know, if $x=\sqrt[]{2+\sqrt{2}}$, then, $x^2=2+\sqrt{2}$
now, if $x=\... | In general, the function $f(x)=\displaystyle\bigg(\small\sqrt{\normalsize x+\small\sqrt{\normalsize x+\sqrt{ x+\sqrt{x}}}}\;\normalsize\bigg)$ is:
$0.5(1+\sqrt[]{1+4x})$
Ref: Math-Integration of nested square roots of x
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/186652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 5
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A cube is divided into two cuboids A cube is divided into two cuboids. The surfaces of those cuboids are in the ratio $7: 5$. Calculate the ratio of the volumes.
How can I calculate this?
| Suppose the original cube had side $a$. And let dimensions of the resulting cuboids be $a \times a \times b$ and $a \times a \times c$, where $a = b+c$.
The surface areas of the resulting cuboids is $2\left(a \cdot a + a \cdot b + a \cdot b\right) = 2 a(a+2 b)$ and $2a(a+2 c)$, and volumes $a^2 b$ and $a^2 c$ respectiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Min/Max of $f_{n}(x)=\lim_{n\to\infty}\frac{(n^{x+1}+1^{x})(n^{x+1}+2^{x})\cdots(n^{x+1}+n^{x})}{(n^{x+1}-1^{x})(n^{x+1}-2^{x})\cdots(n^{x+1}-n^{x})}$ Let's consider the function $f_{n}(x)$ with $x>0$ defined as:
$$f_{n}(x)=\lim_{n\to\infty}\frac{(n^{x+1}+1^{x})(n^{x+1}+2^{x})\cdots(n^{x+1}+n^{x})}{(n^{x+1}-1^{x})(n^{x... | First write
\begin{align*}
\log f_n(x) &= \sum_{j=1}^n \log(n^{x+1}+j^x) - \sum_{j=1}^n \log(n^{x+1}-j^x) \\
&= \sum_{j=1}^n \log\bigg( 1+\frac{j^x}{n^{x+1}} \bigg) + \sum_{j=1}^n \log\bigg( 1- \frac{j^x}{n^{x+1}} \bigg)^{-1}.
\end{align*}
The following inequalities are valid for all $0<y<\frac12$:
\begin{align*}
y-y^2... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why $ \lim_{n \rightarrow \infty}\frac{(5n^3-3n^2+7)(n+1)^n}{n^{n+1}(n+1)^2} =5e$? I have a small exercise and I don’t know who to get the result.
The exercise is: $$ \lim_{n \rightarrow \infty}\frac{(5n^3-3n^2+7)(n+1)^n}{n^{n+1}(n+1)^2} $$
I did following transformations:
$$
\frac{(5n^3-3n^2+7)(n+1)^{n-2}}{n^{n+1}} ... | I would recommend rearranging as
$$
\frac{(5n^3-3n^2+7)(n+1)^n}{n^{n+1}(n+1)^2} = \frac{5n^3-3n^2+7}{n(n + 1)^2}\frac{(n+1)^n}{n^n} = \frac{5n^3-3n^2+7}{n(n + 1)^2} \left ( 1 + \frac{1}{n} \right )^n
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/187768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Permutation across matrices. Matrices may be used to permute the order of elements in a set. For example:
$$
\begin{bmatrix}
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\times
\begin{bmatrix}
x \\
y \\
z \\
... | When you multiply by a permutation matrix on the left, you permute rows (those become single entries in a column vector). When you multiply by a permutation matrix on the right, you permute columns. Here you need to permute rows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/187828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to find $ \cot^2 a + \frac 1{\cot^2 a}$ if $\cot a + \frac 1{\cot a} = 1$ If $ \cot a + \frac 1 {\cot a} = 1 $, then what is $ \cot^2 a + \frac 1{\cot^2 a}$?
the answer is given as $-1$ in my book, but how do you arrive at this conclusion?
| Taking $x=\cot a$, $x+\frac{1}{x}=1\implies x^2+\frac{1}{x^2}=(x+\frac{1}{x})^2-2x\frac{1}{x}=1-2=-1$
Alternatively, $x+\frac{1}{x}=1\implies x^2-x+1=0$
$x^2-x+1=0\implies x^3+1=0$
So,
$x^{3m}+(\frac{1}{x})^{3m}=(x^3)^m+\frac{1}{(x^3)^m}=2(-1)^m$
$x^{3m+1}+(\frac{1}{x})^{3m+1}=(x^3)^m\cdot x+\frac{1}{(x^3)^m\cdot x}=(-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
What is the probability that $XYZ$ is divisible by $5$? A solution of $X + Y + Z = 20$ in non-negative integers is chosen at random. What is the probability that $XYZ$ is divisible by $5$?
Edit:
This happens to be an exam question. So I can't use calculators or computers and have to get the answer in less than 20 minut... | There are two questions here:
*
*How many triples of numbers $(X,Y,Z)$ add to 20? We can call this $A$.
*How many of these are divisible by 5? We can call this $B$.
The answer will then be $B/A$.
First we calculate $A$. Let us write $C(x)$ for the total number of ways of choosing $(X,Y,Z)$ so that $X=x$ and $X+Y+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/190869",
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"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 1
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Evaluating $\int\frac{\cos^{n-1}\frac{x+a}{2}}{\sin^{n+1}\frac{x-a}{2}}\;dx$ Please help me to evaluate the following integral:
$$\int\dfrac{\cos^{n-1}\dfrac{x+a}{2}}{\sin^{n+1}\dfrac{x-a}{2}}\;dx$$
| $$\int\frac{\cos^{n-1}\frac{x+a}{2}}{\sin^{n+1}\frac{x-a}{2}}dx=\int\frac{\cos^{n-1}\frac{x+a}{2}}{\sin^{n-1}\frac{x-a}{2}}\cdot\frac{dx}{\sin^2\frac{x-a}{2}}=\left|\frac{\cos\frac{x+a}{2}}{\sin\frac{x-a}{2}}=t\Rightarrow\frac{dx}{\sin^2\frac{x-a}{2}}=-\frac{2dt}{\cos a}\right|=-\frac{2}{\cos a}\int{t^{n-1}dt}=-\frac{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Prove that $ \frac{1}{2}\cot^{-1}\frac{2\sqrt[3]{4}+1}{\sqrt{3}}+\frac{1}{3}\tan^{-1}\frac{\sqrt[3]{4}+1}{\sqrt{3}}=\dfrac{\pi}{6}. $ Prove that $$ \frac{1}{2}\cot^{-1}\frac{2\sqrt[3]{4}+1}{\sqrt{3}}+\frac{1}{3}\tan^{-1}\frac{\sqrt[3]{4}+1}{\sqrt{3}}=\dfrac{\pi}{6}. $$
| Use the following identities:
$$\arctan a \pm\arctan b=\arctan\left(\frac{a\pm b}{1\mp ab}\right)$$
$$\arctan a+\mathrm{arccot} \hspace{2pt}a=\frac{\pi}{2},\hspace{10 pt}\forall a>0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/193128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Factoring the expression $x^6 + 64$ Alright, so apparently I've factored this out wrong...
$x^6 + 64 =$
$x^6 + 2^6$
Then I continued, using $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ to get ...
$$(x^2)^3 + 64 =
(x^2)^3 + 4^3 =
(x^2 + 4)(x^4 - 4x^2 + 16)$$
How is this incorrect?
| This is correct but you are probably asked to continue with the identity
$$
x^4-4x^2+16=x^4+8x^2+16-12x^2=(x^2+4)^2-12x^2=a^2-b^2,
$$
for some $a$ and $b$ I will let you discover. The final factorisation of $x^6+64$ over the field $\mathbb R$ is the product of three polynomials $x^2+px+q$ with $p^2\lt4q$.
Recall that... | {
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"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Expressing in the form $A \sin(x + c)$
Express in the form $A\sin(x+c)$
a) $\sin x+\sqrt3\cos x$; b) $\sin x-\cos x$
sol: a) $A=\sqrt{1+3}=2$, $\tan c=\frac{\sqrt 3}1$, $c=\frac\pi3$. So $\sin x+\sqrt3\cos x=2\sin(x+\frac\pi3)$
b) $\sqrt 2\sin(x-\frac\pi4)$
Can someone please explain the method used in the provi... | Using the appropriate formula for $\sin$ you have $A \sin(x+c) = A \sin x \cos c + A \cos x \sin c$. You need to determine $A,c$ so the formula holds true for a), b).
Equating $A \sin x \cos c + A \cos x \sin c = \sin x + \sqrt{3} \cos x$ gives $A \cos c = 1$, $A \sin c = \sqrt{3}$. This gives $\tan c = \frac{A \sin c}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
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Prove $ 1 + 2 + 4 + 8 + \dots = -1$
Possible Duplicate:
Infinity = -1 paradox
I was told by a friend that $1 + 2 + 4 + 8 + \dots$ equaled negative one. When I asked for an explanation, he said:
Do I have to?
Okay so, Let $x = 1+2+4+8+\dots$
$2x-x=x$
$2(1+2+4+8+\dots) - (1+2+4+8+\dots) = (1+2+4+8+\dots)$
Therefore, ... | What is wrong is that ``$1+2+4+8+\cdots$'' is not a number, and so you cannot treat it like one.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/193872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 2
} |
How to show that that $\sum_{n=1}^{\infty}\left( \frac{1}{3n-1} + \frac{1}{3n-2}- \frac{2}{3n}\right)= \ln\left(3\right)$? $$
\mbox{How to show that that}\qquad
\sum_{n = 1}^{\infty}\left({1 \over 3n - 1} + {1 \over 3n - 2} - {2 \over 3n}\right)
=
\ln\left(3\right)\ {\large ?}
$$
$$
\mbox{or}\quad
1 + \frac{1}{2} -\fra... | Let $s_N=\sum_{n=1}^N \frac1n$ be the $n$th partial sum of the harmonic series.
Then
$$\sum_{n=1}^N(\frac1{3n-1}+\frac1{3n-2}-\frac2{3n})=\sum_{n=1}^N(\frac1{3n-1}+\frac1{3n-2}+\frac1{3n})-\sum_{n=1}^N\frac1n=s_{3N}-s_N.$$
It is well-known that $s_N=\ln N +\gamma+O\left(\frac1N\right)$, hence
$$\sum_{n=1}^N(\frac1{3n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/197595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
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By using properties of determinants show that determinant is equal to $(1+a^2+b^2)^3$ $$\begin{vmatrix}1+a^2-b^2&2ab&-2b\\
2ab&1-a^2+b^2&2a\\
2b&-2a&1-a^2-b^2\end{vmatrix}=(1+a^2+b^2)^3$$
I have been trying to solve the above determinant. But unfortunately my answer is always coming as:
$$1+3a^2+3a^4+a^6+3a^2b^4+3b^2+4... | $$\begin{vmatrix}1+a^2-b^2&2ab&-2b\\
2ab&1-a^2+b^2&2a\\
2b&-2a&1-a^2-b^2\end{vmatrix}$$
$$=\begin{vmatrix}1+a^2-b^2&2ab&-2b\\
2ab-a(2b)&1-a^2+b^2-a(-2a)&2a-a(1-a^2-b^2)\\
2b&-2a&1-a^2-b^2\end{vmatrix}$$ (applying $R'_2=R_2-aR_3$)
$$=\begin{vmatrix}1+a^2-b^2&2ab&-2b\\
0&1+a^2+b^2&a(1+a^2+b^2)\\
2b&-2a&1-a^2-b^2\end{vma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/197990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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the least possible value for :$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $ If we know that for every $a,b,c>0$ ,how we can find the least possible value for :
$$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $$
| I would come up with one line proof starting from the trivial fact that $\left\lfloor x \right\rfloor>x-1$. Then
$$\lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor>\frac{a}{c}+\frac{c}{a}+\frac{b}{a}+\frac{a}{b}+\frac{c}{b}+\frac{b}{c}-3\ge3.$$
Since the left side is an integer,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Solve the differential equation using Taylor-series expansion Solve the differential equation using Taylor-series expansion:
$$
\frac{dy}{dx} = x + y + xy \\
y (0) = 1
$$
to get value of $y$ at
$x = 0.1$ and $x = 0.5$. Use terms through $x^5$.
| Suppose that $y$ has the Taylor series expansion about $x=0$ given by
$$y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+\cdots.$$
Because $y(0)=1$, we have $a_0=1$.
Differentiate. We get
$$\frac{dy}{dx}=a_1+2a_2x+3a_3x^2+4a_4x^3+5a_5x^4+\cdots.\tag{$1$}$$
Also,
$$\begin{align}x+y+xy&=x+(1+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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Inequality. $2(x^2+y^2+z^2)^2 \geq 3(x^2y^2+y^2z^2+z^2x^2)+3(x^3y+y^3z+z^3x)$ How do I prove that :
$$2(x^2+y^2+z^2)^2 \geq 3(x^2y^2+y^2z^2+z^2x^2)+3(x^3y+y^3z+z^3x).$$
It seems to be easy , but I have no idea:)
thanks :)
| I assume that $x,y,z>0$, even if not explicit from your post. Tell me if I'm wrong.
Expand the $\operatorname{LHS}$. Then we want to prove that
$$2x^4+2y^4+2z^4+4x^2y^2+4y^2z^2+4z^2x^2\geq 3(x^2y^2+y^2z^2+z^2x^2)+3(x^3y+y^3z+z^3x).$$
It is sufficient to estabilish
$$2x^4+2y^4+2z^4+x^2y^2+y^2z^2+z^2x^2\geq 3(x^3y+y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/202183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\forall n \in \mathbb{N^*}$, prove that $|{\frac{\sqrt{n^2+2}}{2n} - \frac{1}{2}}| < \frac{1}{2n^2}$ Let $$x_n = \frac{\sqrt{n^2+2}}{2n},n \in \mathbb{N^*} \ldots$$
Prove that $$|{x_n - \frac{1}{2}}| < \frac{1}{2n^2}$$
Indication: Use the relationship: $\sqrt{1+\gamma} < 1 + \frac{\gamma}{2}$, $ \forall \gamma \in \ma... | First of all we note that $\left|{\frac{\sqrt{n^2+2}}{2n} - \frac{1}{2}}\right| = {\frac{\sqrt{n^2+2}}{2n} - \frac{1}{2}}$ because ${\frac{\sqrt{n^2+2}}{2n} - \frac{1}{2}}>0$. In fact $\frac{\sqrt{n^2+2}}{2n}>\frac{1}{2} \Rightarrow \frac{n^2+2}{n^2}>1 \Rightarrow 1+\frac{2}{n^2}>1$.
Now we note that
$\frac{\sqrt{n^2+2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/202404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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How do we solve the equation? How do we solve the following equation in the set of real numbers?
$$(x+1)\cdot \sqrt{x+2} + (x+6)\cdot \sqrt{x+7}=(x+3)\cdot (x+4).$$
I wrote the given equation has the form
\begin{equation*}
(x+1)(\sqrt{x + 2} - 2) + (x + 6)(\sqrt{x+7} - 3) = (x-2)(x+4)
\end{equation*}
This equation i... | try squaring everything and expanding then reducing ...
| {
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"url": "https://math.stackexchange.com/questions/202670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How do I integrate $\sqrt{x^2+81}$ using trig substitution? How do I integrate $\sqrt{x^2+81}$ using trig substitution? Please be as specific as possible, thank you!
| Jonathan's answer is good, but he left a lot of the work to you when evaluating
$$81 \int \sec^3 t \ dt$$
So, I'll expand on that since it is frequently proved in elementary calculus courses and is a key part of this problem.
Preword: when you see integrals of the form $$\sqrt{a^2+x^2}$$
the substitution $x = a \tan \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/203727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Ideas on the ways to integrate $\int \tan^2( x)\sec^3( x) dx$ I would proceed by thus , let $y = [\sec (x)]^2 $
then
$$dy = 2 \cdot \sec(x) \cdot \sec(x) \cdot \tan(x) \cdot dx = 2 \cdot ( \sec (x))^2 \cdot \tan(x) \cdot dx $$
so,
$$
2 \tan^2(x) \sec^2 (x) dx = \sec(x) \cdot \tan(x) \cdot dy = y(y-1)^\frac{1}{2} \cdo... | The answer for $\int \tan^2 (x)\sec^3 (x) dx$:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/203861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Solving a series $n(1 + n + n^2 + n^3 + n^4 +.......n^{n-1})$ I'm trying to sum the following series?
$n(1 + n + n^2 + n^3 + n^4 +.......n^{n-1})$
Do you have any ideas?
| Let $ n + n^2 + n^3 + n^4 + \cdot\cdot\cdot + n^n = Sum $
Then,
$ 1 + n + n^2 + n^3 + n^4 + \cdot\cdot\cdot + n^n = Sum + 1$
$ n \times (1 + n + n^2 + n^3 + n^4 + \cdot\cdot\cdot + n^n ) = n \times (Sum + 1)$
$ n + n^2 + n^3 + n^4 + \cdot\cdot\cdot + n^n + n^{n+1} = n\times Sum + n$
$ (n + n^2 + n^3 + n^4 + \cdo... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Describe all solutions of the linear system system Describe all solutions of the system (the last column is the augmented column)
$-x_1 +2x_2 + x_3 + 4x_4 = 0$
$2x_1 + x_2 -x_3 +x_4 = 1$
$\pmatrix{-1&2&1&4&0 \\
2&1&-1&1&1
}
\sim \pmatrix{1&-2&-1&-4&0 \\
0&5&1&9&1}
\sim \pmatrix{1&0 & -\dfrac{3}{... | You need to be careful scaling.
You have the equations:
$$-x_1+\frac{3}{5}x_3 + \frac{2}{5}x_4 = -\frac{2}{5}, \ \ \ 5 x_2 + x_3 + 9x_4 = 1$$
which can be written as
$$x_1 = \frac{3}{5}x_3 + \frac{2}{5}x_4 + \frac{2}{5}, \ \ \ x_2 = -\frac{1}{5} x_3 - \frac{9}{5} x_4 + \frac{1}{5}$$
Thus all the solutions are:
$$x = \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Proving an inequality concerning arbitary complex numbers $\def\abs#1{\left|#1\right|}$If $y$ and $z$ are any complex numbers then prove that
\[
2 \abs{y+z}\ge \bigl(\abs y + \abs z\bigr)
\abs{\frac y{\abs y} + \frac z{\abs z}} \]
| $\def\abs#1{\left|#1\right|}$If $y=r(\cos A+i\sin A), z=R(\cos B+i\sin B)$
$|y+z|=\sqrt{r^2+R^2+2Rr\cos(A-B)}$
$$ \abs{\frac y{\abs y} + \frac z{\abs z}}=\sqrt{2+2\cos(A-B)}$$
$2|y+z|$ will be $\ge (|y|+|z|)(\abs{\frac y{\abs y} + \frac z{\abs z}}) $
if $2\sqrt{r^2+R^2+2Rr\cos(A-B)} \ge (r+R)\sqrt{2+2\cos(A-B)}$
if $4(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/210974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Coefficent of an equation I am suppose to find the coefficent of x^2 in this equation
after doing the calculation I ended up with this
but that is the wrong answer what is that I am missing?
| You have $f(x)=x^2+bx+c$, and you want $f\big(f(x)\big)$. As you say, this is $$\left(x^2+bx+c\right)^2+b\left(x^2+bx+c\right)+c\;.\tag{1}$$
Now
$$\begin{align*}
\left(x^2+bx+c\right)^2=x^4+2bx^3+\left(b^2+2c\right)x^2+2bcx+c^2\;,
\end{align*}$$
so $(1)$ becomes
$$x^4+2bx^3+\left(b^2+2c+b\right)x^2+\left(2bc+b^2\right)... | {
"language": "en",
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In how many ways can I have $1.50 using exactly 50 Coins? In how many ways can I have $1.50 using exactly 50 Coins? The coins may be pennies (1 cent), nickels (5 cents), dimes (10 cents) or quarters (25 cents).
| Suppose that you use $p$ pennies, $n$ nickels, $d$ dimes, and $q$ quarters; then $p,n,d$, and $q$ must satisfy the system
$$\left\{\begin{align*}
&p+n+d+q=50\\
&p+5n+10d+25q=150\;.
\end{align*}\right.\tag{1}$$
Thus, you want to count the solutions to $(1)$ in non-negative integers.
The problem is small enough that you ... | {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 5
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rotation of conic sections In the discriminant test of conic sections(rotations), why we're checking with $B^2-4AC$.
How $B^2-4AC=B'^2-4A'C'$, where $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ is changed to $A'x^2+C'y^2+D'x+E'y+F'=0$ using rotations by angle alpha.
| We can find from here, the condition for the general 2nd degree equation to represent a pair of straight lines.
If that condition is not satisfied we can apply the following logic.
A conic is the locus of a point $(x,y)$ which maintains constant ratio(called eccentricity$(e)$) of the distances from a fixed point(called... | {
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"source": "stackexchange",
"question_score": "2",
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Invertible Matrix Let $A$ be the matrix
$$ A=\left(\begin{array}{cccc}1&0&1&2\\2&3&\beta&4\\4&0&-\beta&-8\\ \beta&0&\beta&\beta \end{array}\right). $$
For what values of $\beta$ is the matrix invertible?
| Compute the determinat of $A$ with laplace expansion expanding at second column it will ease the computations because there are there many zeros
$|A|=(-1)^{2+2}3 \left|\begin{array}{ccc}1&1&2\\4&-\beta&-8\\ \beta&\beta&\beta \end{array}\right|$
For $\beta=0 \Rightarrow |A|=0$
Now for $\beta \not =0$
$$|A| \not= 0 \Lef... | {
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"question_score": "2",
"answer_count": 4,
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Integrating $\frac{\log(1+x)}{1+x^2}$
Possible Duplicate:
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$
I am a bit stuck here in evaluating the following integral:$$\int_{0}^{1}\frac{\log(1+x)}{1+x^2}\,\mathrm dx$$.Your help is appreciated.
| A thought: Notice $$\int\limits_0^1 \frac{\log(1 + x)dx}{1 + x^2} = \int\limits_0^1 \frac{\log(1 + x)dx}{1 + 2x -2x + x^2} = \int\limits_0^1 \frac{\log(1 + x)dx}{(x+1)^2 - 2x } \int\limits_0^1 \frac{\log(1 + x)dx}{(x+1)^2 - 2(x+1) + 2}$$ Now put $u = x + 1$ and so youll have
$$\int\limits_1^2 \frac{\log(u)du}{u^2 - 2u ... | {
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What is the range of the function $f:[-1,1] \rightarrow \mathbb{R}$ defined by $f(x) = 12x^2 + 5x\sqrt{1-x^2} - 10$? I was navigating through math exercises about functions and I got with this question
If $f:[-1,1] \rightarrow \mathbb{R}$ defined by $f(x) = 12x^2 + 5x\sqrt{1-x^2} - 10$.
Give the range of $f(x)$ in the... | Finding the derivative and solving $f'(x)=0$ may be unpleasant. So it is natural to let $x=\cos t$ where $0\le t\le \pi$. Our function is then $12\cos^2 t+5\cos t\sin t-10$. If we are in a calculus mood, differentiate.
But if we are in a trigonometric mood, we can use the identities $\cos 2t=2\cos^2 t-1$ and $\sin 2t=... | {
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"source": "stackexchange",
"question_score": "3",
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Prove the following relation: I must prove the relation $$\sum_{k=0}^{n+1}\binom{n+k+1}{k}\frac1{2^k}=2\sum_{k=0}^n\binom{n+k}{k}\frac1{2^k}.$$
I got this far before I got stuck:
$\begin{eqnarray*}
\sum_{k=0}^{n+1}\binom{n+k+1}{k}\frac1{2^k} & = & \sum_{k=0}^{n+1}\left\{\binom{n+k}{k}+\binom{n+k}{k-1}\right\}\frac1{2^k... | By way of enrichment here is another algebraic proof using basic
complex variables.
Suppose we are trying to show that
$$\sum_{k=0}^{n+1} {n+1+k\choose k} \frac{1}{2^k}
= 2 \sum_{k=0}^n {n+k\choose k} \frac{1}{2^k}.$$
Introduce the integral representations
$${n+1+k\choose k}
= \frac{1}{2\pi i}
\int_{|z|=\epsilon... | {
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"source": "stackexchange",
"question_score": "4",
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The quadratic form $x^2 + ny^2$ via prime factors Elementary algebra shows that the product of two numbers in the form $x^2 + ny^2$ again has the same form, since if $p = (a^2 + nb^2)$ and $q = (c^2 + nd^2)$,
$$pq = (a^2 + nb^2)(c^2 + nd^2) = (ac \pm nbd)^2 + n(ad \mp bc)^2$$
My question is: Assuming that a number $z$... | Well, no. It is a fair question, though. Even with class number one, we can begin with $1 + 3 = 4,$ although $x^2 + 3 y^2$ does not represent $2.$ The way Dickson would have talked about this is the imprimitive form of the same discriminant, namely $2 x^2 + 2 x y + 2 y^2.$
Staying with one class per genus, we have $1 +... | {
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"source": "stackexchange",
"question_score": "5",
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Proving with factorials Let x and y be the postive integers.
Show that :
$\displaystyle\frac{(x + y)!}{ (x + y)^{(x + y)}} < \frac{x! y!}{ (x^x + y^y)}$
Are there any identities we can use to easily prove this?
| Edit: The statement in itself is not correct: If $x=y=1$, both sides are $\frac{1}{2}$.
Nevertheless, the statement is correct if $x > 1$ or $y > 1$. I'll first prove it for $x,y$ both $>1$, and then for the case where one of them is 1.
I will first assume that $x > 1$, $y > 1$. First, note that $\binom{x+y}{x} = \frac... | {
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"source": "stackexchange",
"question_score": "2",
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How to integrate $\int 1/(x^7 -x) dx$? How should I proceed about this integral?
$$\int {1/(x^7 -x)} dx$$
I've tried integration by parts or substitution but I can't seem to solve it. Can I have some hints on how should I get started?
These are some of the things I've tried:
IBP: $u = \frac {1}{x^6-1}$, $du = \frac {-... | As in this example it is useful to extract something of the form
$$u=x^a\pm\frac{1}{x^a}$$
$$\int\frac{dx}{x^{7}-x}=\int\frac{1}{x^{4}}\frac{dx}{x^{3}-\frac{1}{x^{3}}}=\int\frac{\left(x^{2}+\frac{1}{x^{4}}\right)dx}{x^{3}-\frac{1}{x^{3}}}-\int\frac{x^{2}dx}{x^{3}-\frac{1}{x^{3}}}=I_1-I_2$$
Now in $I_1$ we may let
$$u=x... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
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How can I calculate the limit of exponential divided by factorial? I suspect this limit is 0, but how can I prove it?
$$\lim_{n \to +\infty} \frac{2^{n}}{n!}$$
| To prove $\lim_{n\to \infty} \,\frac{n!}{a^n} = +\infty$ it is suffice to prove that for all $k>0$ it is possible $\frac{n!}{a^n} > K$ for almost $n$, that is $n!>K a^n$. This means
$1\cdot 2\cdot\ldots\cdot\frac{n}{2}\cdot\ldots \cdot n > K \cdot a\cdot \stackrel{\stackrel{n}{\smile}}{\ldots}\cdot a,$
that is
$\frac{n... | {
"language": "en",
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"source": "stackexchange",
"question_score": "9",
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We define a sequence of rational numbers {$a_n$} by putting $a_1=3$ and $a_{n+1}=4-\frac{2}{a_n}$ for all natural numbers. Put $\alpha = 2 + \sqrt{2}$ We define a sequence of rational numbers {$a_n$} by putting $a_1=3$ and $a_{n+1}=4-\frac{2}{a_n}$ for all $n \in \mathbb N$. Put $\alpha = 2 + \sqrt{2}$
(a) Prove by ind... | Certainly $a_1 \le 3 \lt 4$, since $a_1=3$.
Suppose that for the specific integer $k$, we know that $3\le a_k \lt 4$.
We want to show that similar inequalities hold for the "next" term $a_{k+1}$. That is, we want to show that $3\le a_{k+1}\lt 4$.
There are two inequalities to prove. Let us deal with them separately. ... | {
"language": "en",
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"source": "stackexchange",
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$2n+1$ and $n^2+1$ are always coprime or their gcd is $5$ Using a spreadsheet, it can be inferred that when $n≡2[5]$, then $\gcd(n^2+1,2n+1)=5$, else $\gcd(n^2+1,2n+1)=1$.
Indeed, when $n≡2[5]$, $n^2+1$ and $2n+1$ can easily be shown to be multiples of $5$, so their gcd is at least $5$. But then, I can't see how to com... | Clearly when $n=0$ the polynomials are relatively prime (gcd = 1).
notice that $2n+1$ is always odd, while $n^2 + 1$ is always even. Thus the gcd can never contain $2$ as a factor.
Let $d | n^2 + 1$ and $d | 2n + 1$, then [because $a|b$ implies $a|kb$] we have
*
*$d|2n^2 + 2$
*$d|2n^2 + n$
and [because $a|b$ an... | {
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"source": "stackexchange",
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Why ${ \sum\limits_{n=1}^{\infty} \frac{1}{n} }$ is divergent , but ${ \sum\limits_{n=1}^{\infty} \frac{1}{n^2} }$ is convergent? I don't understand why ${ \displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n} }$ is divergent, but ${ \displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n^2} }$ is convergent and its limit i... | The following example may be easier to grasp. Consider the series
$$1+\frac{1}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\cdots. $$
So we have $2$ copies of $\frac{1}{2}$, $4$ copies of $\frac{1}{4}$, $8$ copies of $\frac{1}{8}$, $16$ copi... | {
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Prove that if $ |a_n-a_{n-1}| < \frac{1}{2^{n+1} }$ and $a_0=\frac12$, then $\{a_n\}$ converges to $0I try to solve this question but I don't know how.
given $ a_0 = \frac12 $ and for each $n\geq 1$:
$$ |a_n-a_{n-1}| < \frac{1}{2^{n+1}} $$
show that $\{a_n\}$ converges and the limit is $a$ such that $0<a<1$
Update (Edi... | Hint: $\vert a_n - a_0 \vert = \vert a_n - a_{n-1} + a_{n-1} - \dots - a_1 + a_1 - a_0 \vert \leq \sum_{k=1}^n \vert a_k - a_{k-1} \vert$
| {
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"source": "stackexchange",
"question_score": "2",
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How to convert this into a single closed form? I have a loop that iterates a variable $i$ from 0 to some boundary $b$. Let $p=4i+4$. Each iteration of the loop I add $(p^3+5p)/3+2$ to a sum variable $s$ which is initially set to 0.
Question: Instead of this loop, how can I convert this all into a closed form expression... | In the end of the loop, if I understand correctly, you have:
$$\begin{align*}s&=\sum_{i=0}^b \left(\frac{(4i+4)^3+5(4i+4)}{3}+2\right)=\sum_{i=0}^b \left(4\frac{16i^3+48i^2+53i+21}{3}+2\right)\\&=\frac{64}{3}\sum_{i=0}^b i^3+64\sum_{i=0}^b i^2+\frac{212}{3}\sum_{i=0}^b i+30\sum_{i=0}^b 1
\end{align*}$$
Using the formul... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Numerical linear algebra (pseudoinverse of a matrix) Let $A$ be the matrix:
$$\left(\begin{matrix}
\alpha I_{n} \\
\beta I_{n}
\end{matrix}\right)$$
where $\alpha,\beta\in\Bbb C$ are not both zero. Derive (a) the (reduced) QR factorization of $A$ and (b) the pseudoinverse of $A$.
Any help for the second question a... | Example $\mathbf{Q} \mathbf{R}$
$$
\begin{align}
\mathbf{A} &= \mathbf{Q} \, \mathbf{R} \\
% A
\left[
\begin{array}{ccc}
a & 0 & 0 \\
0 & a & 0 \\
0 & 0 & a \\
b & 0 & 0 \\
0 & b & 0 \\
0 & 0 & b \\
\end{array}
\right]
%
&=
% Q
\left( a^{2}+b^{2} \right)^{-\frac{1}{2}}
\left[
\begin{array}{ccc}
a & 0 & 0 \\
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do you compute the limit with multiple variables without fail? $$\lim\limits_{(x,y)\rightarrow (0,0)} \dfrac{(x^3-y^3) \sin (2x^2+3y^2)}{x^2+2y^2}$$
I don't know what to do doesn't it give always $0$?
Whether $x=0$ $x=y$ or $y=0$ or $x=y^2$ it always give $0$ since it goes to $(0,0)$
| HINT
Note that $$-2 \leq \dfrac{\sin(2x^2 + 3y^2)}{x^2 + 2y^2} \leq 2$$
Hence, $$(x^3 - y^3)\dfrac{\sin(2x^2 + 3y^2)}{x^2 + 2y^2} \in \left[-2\vert(x^3-y^3) \vert, 2\vert(x^3-y^3)\vert \right]$$
| {
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"source": "stackexchange",
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How to multiply polynominals I can't figure out how to multiply these polynominals $$(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)$$
I tried multiplying like this
$$(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)$$
$$3x^4-7x^3+(5x^2)(2x^2)(-6x^2)+(5x)(7x)(-4x)+(8)(9)$$
$$3x^4-7x^3-60x^2-140x+72$$
It says the answer is $$-12x^6-19x... | You have to multiply every term by every other term. A good way to make sure you don't miss any is to use a table.
First, combine like terms within each group of parentheses:
$$(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)\\
=(3x^4-7x^3+5x^2+5x+8)(-4x^2+3x+9)$$
Then form a table and multiply each term by multiplying the coe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/253522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How to find the sum of an alternating series?
Find whether the series diverges and its sum:
$$\sum_{n = 1}^\infty (-1)^{n+1} \frac{3}{5^n}.$$
I found that the series converges using the Alternating Series test because the absolute value of each $n$ decreases while the value of $n$ increases. Then I took the limit a... | Notice that$$(-1)^{n+1}\frac{3}{5^n}=-3\frac{(-1)^{n}}{5^{n}}=-3\left(\frac{-1}{5}\right)^{n}$$
Since $\sum_{k=1}^{\infty}ar^{k}=\frac{ar}{1-r}$ (iff $|r|<1$), $$\sum_{n=1}^{\infty}-3\left(\frac{-1}{5}\right)^{n}=\frac{-3\cdot\frac{-1}{5}}{1-\frac{-1}{5}}=\frac{\frac{3}{5}}{\frac{6}{5}}=\frac{1}{2}$$ and the sum conver... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/253997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Integrating a function over a domain How could you integrate the function $f(x,y) = x^2 + y^2$ over the triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$?
I define the set $D = \{(x,y)\; |\; 0\leq x\leq 1 \text{ and } 0\leq y\leq x\}$ and then calculate
$$\int_0^1 \int_0^x x^2 + y^2 \; \mathrm{d}y \; \mathrm... | I think you have the limits changed and wrong. I think it must be
$$\int_0^1dx\int_0^{1-x}(x^2+y^2)dy=\int_0^1\left.\left(x^2y+\frac{1}{3}y^3\right|_0^{1-x}\right)dx=\int_0^1\left(x^2(1-x)+\frac{1}{3}(1-x)^3\right)dx=$$
$$=\frac{1}{3}-\frac{1}{4}-\frac{1}{12}(0-1)=\frac{2}{12}=\frac{1}{6}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that for any triple $a_{i},a_{j},a_{k} $ are three edge lengths of some triangle $ a_{1},a_{2},....,a_{n}(n\geq 3) $ are positive numbers that :
$(a_{1}+a_{2}+....+a_{n})^2 >\frac{3n-1}{3}(a_{1}^2+a_{2}^2+....+a_{n}^2)$
Prove that for any triple $a_{i},a_{j},a_{k} $ are three edge lengths of some triangle, where ... | Since the condition is symmetric about $a_1,\dots,a_n$, it suffices to show that $a_1+a_2>a_3$.
When $n=3$, by Cauchy-Schwarz inequality,
$$2\left((a_1+a_2)^2+a_3^2\right)\ge \left((a_1+a_2)+a_3\right)^2 \quad{and}\quad a_1^2+a_2^2\ge \frac{1}{2}(a_1+a_2)^2.$$
Combining the two inequalities above with your condition ... | {
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Find the upper and lower limits of $xf(x)$, as $x\rightarrow \infty$ Define $$f(x)=\int_{x}^{x+1}\sin(t^2)dt$$
Find the upper and lower limits $xf(x)$, as $x\rightarrow \infty$.
I find the answer as $+1, -1$ since $|\sin(x)| \le 1$. (Of course I calculated that function)
Is that right or did I miss something?
=======... | Here is a clarified version of your proof. (In fact, it was a part of my original solution, but I modified it by following your idea.)
Step 1. Estimation of $xf(x)$
Let $t = \sqrt{u}$. Then
\begin{align*}
xf(x)
&= x\int_{x}^{x+1} \sin (t^2) \, dt = x \int_{x^2}^{(x+1)^2} \frac{\sin u}{2\sqrt{u}} \, du \\
&= x \left[ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A question regarding case notation Consider the parameter $P$ and a property $S$ associated with this parameter represented as $P_S$. For example, consider that the height of a tree (“H”) is my parameter and the property is the tree’s name. Therefore, $H_{T1}$ refers to the height of the tree named $T1$.
Now, consider ... | You could use $H_{T1,A,B}$, or $H_{T1;A,B}$. Your idea for your function is almost right, but you've included some superfluous information. Try
$H_{T1,A,B,C,D}=\cases{a, \text{if } A=1,B=1,∀C,D,\\b, \text{if }A=2,B=2, ∀ C,D,\\c, \text{if } A=3,B=3,C=1,D=2,\\d, \text{if } A=3,B=3,C=1,D=3}$
or use your simplifying trick... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve system of equations? I want to sove the system of equations $$\begin{cases}
x^3 y-y^4=7,\\
x^2 y+2 xy^2+y^3=9.
\end{cases}
$$
I tried divide these two equations we obtain
$$\dfrac{x^3 - y^3}{(x+y)^2 } = \dfrac{7}{9}$$
From here, I don't know how to solve.
| $$\begin{cases}
x^3 y-y^4=7,\\
x^2 y+2 xy^2+y^3=9.
\end{cases}$$
$x^3y-y^4=y(x^3-y^3)$, while $x^2y+2xy^2+y^3=y(x+y)^3$.
We try the transformation $x+y=z$.
Then $y(x^3-y^3)=y(x-y)(x^2+xy+y^2)=y(z-2y)(z^2-xy)=y(z-2y)(z^2-(z-y)y)=$
$-2y^4+3y^3z-3y^2z^2+yz^3=7$
On the other hand, the second equation gives $yz^2=9$, so $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove among two integers $2^n-1$ and $2^n+1$, at most one is prime. For any $n>2$, among $2^n-1$ and $2^n+1$, at most one is prime.
I have no clue :(
| As $(2^n-1),2^n,(2^n+1)$ are three consecutive integers, exactly one of them is divisible by $3$
Now, $(2,3)=1\implies 3\mid(2^n-1)(2^n+1)$
which can also proved as $(2^n-1)(2^n+1)=4^n-1^n$ which is divisible by $4-1=3$ as $(a-b)\mid(a^m-b^m)$ where $m$ is any natural number.
So, exactly one of them is divisible by $3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Compute $\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$ Compute the limit
$$\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$$
| Let's suppose that $\ \displaystyle f(n):=\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx\ $ then :
\begin{align}
f(n+1)-f(n)&=\int_0^{\pi} \frac{\sin^2((n+1) x)-\sin^2(n x)}{\sin x} \ dx\\
&=\int_0^{\pi} \frac{\cos(2n x)-\cos(2(n+1) x)}{2\sin x} \ dx\\
&=\int_0^{\pi} \frac{\cos(2n x)(1-\cos(2x))+\sin(2nx)\sin(2x)}{2\sin x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 1
} |
sum of this series: $\sum_{n=1}^{\infty}\frac{1}{4n^2-1}$ I am trying to calculate the sum of this infinite series after having read the series chapter of my textbook: $$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}$$
my steps:
$$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}=\sum_{n=1}^{\infty}\frac{2}{4n^2-1}-\sum_{n=1}^{\infty}\frac{1}... | Note $\frac{1}{4n^2-1}=\frac{1}{(2n+1)(2n-1)}={\frac{1}{2}}\times\frac{(2n+1)-(2n-1)}{(2n+1)(2n-1)}={\frac{1}{2}}\times[\frac{1}{2n-1}-\frac{1}{2n+1}]$ for $n\in\mathbb N.$
Let for $k\in\mathbb N,$ $S_k=\displaystyle\sum_{n=1}^{k}\frac{1}{4n^2-1}$ $\implies S_k={\frac{1}{2}}\displaystyle\sum_{n=1}^{k}\left[\frac{1}{2n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 4
} |
Compute the series Compute the series
$$1)\space\sum_{n=1}^{\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n+1}\right)\frac{1}{n(n+1)}$$
$$2)\space\sum_{n=1}^{\infty}\left(1-\frac{1}{2}+\frac{1}{3}-\cdots(-1)^{n}\frac{1}{n+1}\right)\frac{1}{n(n+1)}$$
| In both of these, we will use the telescoping sum
$$
\begin{align}
\sum_{k=n}^\infty\frac1{k(k+1)}
&=\sum_{k=n}^\infty\frac1k-\frac1{k+1}\\
&=\frac1n
\end{align}
$$
$1)$ Here is one way:
$$
\begin{align}
\sum_{n=1}^\infty\sum_{k=0}^n\frac1{k+1}\frac1{n(n+1)}
&=1+\sum_{n=1}^\infty\sum_{k=1}^n\frac1{k+1}\frac1{n(n+1)}\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
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Fractions in Questions and Answers
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