Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
How do I write a vector as a linear combination of other vectors. Write $\begin{pmatrix} 5 \\ 3 \\15 \end{pmatrix}$ as a linerar combination of the following vectors: $u=\begin{pmatrix} 1 \\ 2 \\5 \end{pmatrix}$, $v=\begin{pmatrix} 3 \\ -4 \\-1 \end{pmatrix}$, $w=\begin{pmatrix} -1 \\ 1 \\1 \end{pmatrix}$.
My atte... | So...you've done the hard part, now finish it off:
$$c_1\begin{pmatrix} 1 \\ 2 \\5 \end{pmatrix} + c_2 \begin{pmatrix} 3 \\ -4 \\-1 \end{pmatrix}+ c_3\begin{pmatrix} -1 \\ 1 \\1 \end{pmatrix} = \begin{pmatrix} 5 \\ 3 \\15 \end{pmatrix}$$
So what have you found to be $c_1, c_2, c_3$?
Fill in the correct values for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/299959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$ Prove by Mathematical Induction . . .
$1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$
I tried solving it, but I got stuck near the end . . .
a. Basis Step:
$(1)(1!) = (1+1)!-1$
$1 = (2\cdot1)-1$
$1 = 1 \checkmark$
b. Inducti... | $$1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$$
then
$$1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!)+(n+1)(n+1)! =(n+1)!-1+(n+1)(n+1)!=$$
$$(n+1)!(n+1+1)-1=(n+2)!-1=((n+1)+1)!-1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/301615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
If $a$ and $b$ are two roots of $x^4 + x^3 - 1 = 0$ prove that $ab$ is a root of $x^6 + x^4 + x^3 - x^2 - 1$. If $a$ and $b$ are two roots of $x^4 + x^3 - 1 = 0$ prove that $ab$ is a root of $x^6 + x^4 + x^3 - x^2 - 1$.
Students and I are unsure how to go about this problem. Also will this be a problem I can solve and ... | There is probably a shorter way, but I think this way sheds some light on why it's true:
Let $a,b,c,d$ be the four roots of $x^4+x^3-1$, so that $x^4+x^3-1=(x-a)(x-b)(x-c)(x-d)$. Now set
$$
g(x) = (x-ab)(x-ac)(x-ad)(x-bc)(x-bd)(x-cd).
$$
This polynomial is symmetric in the roots $a,b,c,d$, and so its coefficients will ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 2
} |
Solve the congruence$x^3+4x+8\equiv{0}\pmod{15}$ Solve (if possible)the congruence involving polynomial
$x^3+4x+8\equiv{0}\pmod{15}$
My work:
Since $15=3\cdot5$, we have
$x^3+4x+8\equiv{0}\pmod{3}$ and $x^3+4x+8\equiv{0}\pmod{5}$
In $\mathbb{Z}_3$,
We have $[0],[1],[2]$
They all dont work
In $\mathbb{Z}_5$,
We have ... | Modulo $3$, there is a solution: $x\equiv 2\pmod{3}$ does work.
But modulo $5$, there is no solution.
So there is no solution modulo $15$. For if $x^3+4x+8\equiv 0\pmod{15}$, then $x^3+4x+8\equiv 0\pmod{5}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/302677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
The remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$ How to find the remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$
| The sum is of the form
$$\sum_{k=0}^{125}((8k+1)^2+(8k+3)^2+(8k+5)^2+(8k+7)^2)+((8.126+1)^2+(8.126+3)^2+(8.126+5)^2)$$
This above one is congruent to(mod 8),
$$(\sum_{k=0}^{125}((1)^2+(3)^2+(5)^2+(7)^2))+(1+3^2+5^2)$$
This is again congruent to,
$$(\sum_{k=0}^{125}(1+1+1+1))+(1+1+1)=126\times 4+3$$
This is congruent t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Proving $4^{47}\equiv 4\pmod{12}$ I know this is a simple exercise, but I was wondering if I can make the following logical jump in my proof:
We see that $4\equiv 4\pmod{12}$ and $4^2\equiv 4\pmod{12}$. Then we can recursively multiply by $4$ to get $4^{47}\equiv 4\pmod{12}$.
| $4^{47}$ and $4$ are divisible by $4$ thus $4^{47}-4$ is a multiple of $4$.
Also, since $4 \equiv 1 \pmod{3}$ we have $4^{46}\equiv 1^{46}\pmod{3}$ and then
$$4^{47} \equiv 4 \pmod 3 \,.$$
Thus $4^{47}-4$ is a multiple of $3$ and $4$, and hence of 12...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/304224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
The area of the triangle with vertices $(3, 2), (3, 8)$, and $(x, y)$ is $24$. What is $x$? The area of the triangle with vertices $(3, 2), (3, 8)$, and $(x, y)$ is $24$. A possible value for $x$ is:
*
*a) $7$
*b) $9$
*c) $11$
*d) $13$
*e) $15$
Please show your work and explain.
| We know the area of a triangle (Article#25) having vertices $(x_i,y_i)$ for $i=1,2,3$ is
$$\frac12\det\begin{pmatrix} x_1 & y_1 & 1\\x_2&y_2&1\\ x_3 & y_3 &1\end{pmatrix}$$
Now, $$\det\begin{pmatrix}x & y &1\\ 3 & 2 & 1 \\ 3 & 8 & 1 \end{pmatrix}$$
$$=\det\begin{pmatrix}x-3 & y-2 &0\\ 3 & 2 & 1 \\ 0 & 6 & 0 \end{pmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prove that $\frac{a}{b^2+5}+ \frac{b}{c^2+5} + \frac{c}{a^2+5} \le \frac 12$ Let $a,b,c>0$ and $a^3+b^3+c^3=3$. Prove that $$\dfrac{a}{b^2+5}+ \dfrac{b}{c^2+5} + \dfrac{c}{a^2+5} \le \dfrac 12$$
I have an ugly solution for this solution.
| Lohwater's "Introduction to inequalities" is a tour de force in proving all sorts of inequalities using mostly elementary means.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/306024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Complex Integration : $\int_1^{1+i}\frac{1}{1+z^2}dz$
Integrate alonf the line segment from $z=1$ to $z=1+i$ :
$$\int_1^{1+i}\frac{1}{1+z^2}dz$$
If I integrate, it is just the identity $tan^{-1}z$, but the answer to this question is $$\frac{\pi}{4}-\frac{1}{2}\arctan2+\frac{i}{4}\log5$$ which I don't understand how... | Set $z=1+i t$, $dz = i dt$. Then the integral is
$$\begin{align}i \int_0^1 \frac{dt}{1+(1+i t)^2} &= i \int_0^1 \frac{dt}{2+i 2 t -t^2} \\ &= -i \int_0^1 \frac{dt}{(t-(1+i))(t+(1-i))}\\ &= -\frac{i}{2} \int_0^1 dt \left (\frac{1}{t-(1+i)} - \frac{1}{t+(1-i)} \right ) \\ &= -\frac{i}{2} \left [ \log{ \left ( \frac{t-(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Silly question: Why is $\sqrt{(9x^2)} $ not $3x$? I had to find the derivative of $f(x) = \sqrt{(9x^2)}$. I applied chain rule with the following steps.
Let $f(x)$ be $\sqrt{x}$ and $g(x)$ be $9x^2$
$$ \begin{align} &f'(g(x)) \times g'(x) \\
& = \frac{1}{2\sqrt{(9x^2)}} \times 18x \\
& = \frac{18x}{2\sqrt{9x^2}} \\... | Remember that $\sqrt{x^2}$ is the positive number whose square is $x^2$ (this is the definition of the square root sign). If $x$ is positive, then that number is $x$, otherwise, it is $-x$. For example, $\sqrt{(-3)^2} = \sqrt{9} = 3$, not $-3$. You can always say that $\sqrt{x^2} = \left| x \right|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/308103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
To check the solution via radicals Given equations are
$$x^5+y^5=a \tag 1$$
$$5xy(x^2+xy+y^2)=b \tag 2$$
Is it possible to find $x$ or $y$ via using radicals?
My attempt
$$x^2+y^2=\frac{b}{5xy}-xy$$
$$x^4+y^4=\frac{b^2}{5^2x^2y^2}-x^2y^2-\frac{2b}{5}$$
$$(x^2+y^2)(x^4+y^4)=\left(\frac{b}{5xy}-xy\right)\left(\frac{b... | No. First, notice that
$$ (x+y)^5 = (x^5+y^5) + 5 x y (x^2+xy+y^2) (x+y)$$
So your equations imply
$$(x+y)^5 = a + b(x+y)$$
Putting $s = x+y$, if $x$ and $y$ could be expressed in terms of $a$ and $b$ using radicals, then so could the root of $s^5 -bs -a$. This quintic cannot be solved in radicals. (In fact, this quint... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Formula for $\sum _{i=1}^n (n+1-i) (n-i)$ It is easy to show that
$$\sum _{i=1}^n (n+1-i) (n-i) = n (n-1)+(n-1) (n-2)+...+1 (1-1)=\frac{1}{3} \left(n^3-n\right)$$
using induction. But how do I derive this formula? I couldn't find any substitution to do this.
| You must already know that
$$
\begin{align}
\sum_{i = 1}^n 1 &= n \\
\sum_{i = 1}^n i &= \frac{n(n + 1)}{2} \\
\sum_{i = 1}^n n &= n \cdot \sum_{i = 1}^n 1 \\
& = n^2 \\
\sum_{i = 1}^n i^2 &= \frac{n(n + 1)(2n + 1)}{6} \\
\end{align}
$$
Now, from the given expression;
$$
\begin{align}
\sum_{i=1}^n (n+1-i) (n-i) &= \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$ I need to show that
$$
\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}
$$
I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result which I am supposed to start from. Using a change of variable... | Integrating the integral by parts thrice yields
$$
\begin{aligned}
& \int_{0}^{\infty} \frac{\sin ^{4} x}{x^{4}} d x \\
=& \int_{0}^{\infty} \frac{\left(\sin ^{4} x\right)^{(3)}}{3 \times 2 \cdot x} d x \\
=& \int_{0}^{0} \frac{4 \sin 2 x-8 \sin 4 x}{6 x} d x \\
=& \frac{4}{3}\left(-\int_{0}^{\infty} \frac{\sin 2 x}{x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 6,
"answer_id": 5
} |
Connecting midpoints of sides of a triangle In triangle $\triangle ABC$, $AB=8$, $BC=14$ and $CA=10$. Points $M$, $N$, and $P $ are the midpoints of sides $AB$, $BC$, and $CA$, respectively. If $M$, $N$, and $P$ are connected to form a triangle, what is the perimeter of triangle $\triangle MNP$?
|
$\displaystyle \frac{8}{14}=\frac{4}{MP}\Rightarrow MP=7$
$\displaystyle \frac{14}{8}=\frac{7}{NP}\Rightarrow NP=4$
$\displaystyle\frac{14}{10}=\frac{7}{MN}\Rightarrow MN=5$
Perimeter$=MP+NP+MN=7+4+5=16$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/319945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Evaluating $\int_{0}^{\infty}\sin^3(x)\cos[a\tan(x)]\frac{dx}{x}$ $$I(a)=\int_{0}^{\infty}\sin^3(x)\cos[a\tan(x)]\frac{dx}{x}$$ I'd like to evaluate the integral by differentiating with respect to parameter $a$ but no success yet. Seems impossible. What would the other options?
Edit: A hypothetical closed form solut... | Let us assume that $f(z)$ has a Maclaurin series expansion with real coefficients that converges on the unit circle on the complex plane.
Then
$$ \text{Re} \int_{0}^{\infty} \frac{\sin^{3} x}{x} f(e^{2ix})\, dx = \int_{0}^{\infty} \frac{\sin^3 x}{x} \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} \cos(2nx) \, dx .$$
Now ass... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/322704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 3,
"answer_id": 1
} |
Integrate $\int_{0}^{\pi}{\frac{x\cos{x}}{1+\sin^{2}{x}}dx}$ Integrate
$$\displaystyle \int_{0}^{\pi}{\frac{x\cos{x}}{1+\sin^{2}{x}}dx}$$
| Here's a more elementary way:$$J=\int_0^\pi\frac{x \cos(x) dx}{1+\sin^2(x)}= \int_0^\frac{\pi}{2} \frac{x \cos(x)dx}{1+\sin^2(x)}+\int_0^\frac{\pi}{2} \frac{-x \sin(x)dx}{1+\cos^2(x)}-\frac{\pi}{2} \int_\frac{\pi}{2}^0 \frac{- \sin (x) dx}{1+\cos^2(x)}$$
$$J=I_1-I_2-\frac{\pi^2}{8}$$
$$I_1=\int_0^\frac{\pi}{2} \frac{ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
How to show that $ \frac{\cos x}{1 - \sin x} - \tan x = \sec x$? Question is: verify the identity:
$$
\frac{\cos x}{1 - \sin x} - \tan x = \sec x.
$$
How do I show that the left side equals the right?
I changed $\tan x$ into $\sin x/\cos x$ but didn't get anywhere.
Please help.
| $\frac{\cos x}{1-\sin x} - \tan x = \sec x$
$\displaystyle\frac{\cos x}{1-\sin x} - \frac{\sin x}{\cos x} = \sec x$
$$(\cos^2 x - \sin x (1 - \sin x)) / (1-\sin x) (\cos x) = \sec x$$
$$(\cos^2 x - \sin x + \sin^2 x) / (1-\sin x)\cos x = \sec x$$
$$\begin{array}{lll}
(1 - \sin x) / ((1-\sin x)\cos x) &=& \sec x\\
1/\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/324104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Infinite Series :$ \sum_{n=0}^\infty \frac{\Gamma \left(n+\frac{1}{2} \right)\psi \left(n+\frac{1}{2} \right)}{n! \left(n+\frac{3}{2}\right)^2}$ Prove that:
$$\sum_{n=0}^\infty \frac{\Gamma \left(n+\frac{1}{2} \right)\psi \left(n+\frac{1}{2} \right)}{n! \left(n+\frac{3}{2}\right)^2} = \frac{-\pi^{\frac{3}{2}}}{12}\left... | Begin with
$$\sum_{n=0}^\infty \frac{\Gamma(n+1-y)}{n!}\frac{1}{x+n} = \frac{\pi \Gamma(x)}{\sin(\pi y) \Gamma(x+y)}$$
Differentiating with respect to $x$ gives
$$-\sum_{n=0}^\infty \frac{\Gamma(n+1-y)}{n!}\frac{1}{(x+n)^2} = \frac{\pi \Gamma(x)}{\sin(\pi y) \Gamma(x+y)} \left\{ \psi(x)-\psi(x+y)\right\}$$
Now, differe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/324488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 1
} |
Prove that a set of matrices is a subspace I'm self studying linear algebra and now I'm starting with proofs and so on. I found this exercise and this is the way I prove it. I think it's correct but I'm not sure I mean, what do you think?
Is the set of matrices
$
\begin{pmatrix}
x && x+y \\
x-y && y
\end{pmatrix... | Note
$$
\begin{pmatrix}
x && x+y \\
x-y && y
\end{pmatrix}
=
x \begin{pmatrix}
1 && 1 \\
1 && 0
\end{pmatrix}
+
y \begin{pmatrix}
0 && 1 \\
-1 && 1
\end{pmatrix}
$$
so it is the subspace generated by the two matrices
$$
\begin{pmatrix}
1 && 1 \\
1 && 0
\end{pmatrix},
\qquad
\begin{pmatrix}
0 && 1 \\
-1 && 1
\end{pmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/325020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Proof: Two non identical circles have at most 2 same points I'm struggeling with an analytic proof for the fact, that two different circles have at most 2 same points. (I try to solve it analytical, because geometrical I already prooved it).
I tried to start with the equations
$r_1^2=(x-a_1)^2+(y-b_1)^2$ and $r_2^2=(x-... | A circle is defined as the locus of a degree 2 polynomial.
The intersection of two circles is then, the roots of a degree 2 polynomial
There are at most $n$ roots of a degree $n$ polynomial
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/326654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
A question about an equilateral triangle Suppose that $\triangle ABC$ is an equilateral triangle. Let $D$ be a point inside the triangle so that $\overline{DA}=13$, $\overline{DB}=12$, and $\overline{DC}=5$. Find the length of $\overline{AB}$.
| The Wikipedia article on equilateral triangles quotes the following theorem from "Curious Properties of the Circumcircle and Incircle of an Equilateral Triangle," by Prithwijit De (http://ms.appliedprobability.org/data/files/Abstracts%2041/41-1-7.pdf):
Theorem: Let $ABC$ be an equilateral triangle with side $s$, and le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/330333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
triple integral to calculate volume Let $V$ be the volume of the $3$-dimensional structure bounded by the paraboloid $z=1−x^2−y^2$, planes $x\ge0, y\ge0$ and $z\ge0$ and by the cylinder $x^2+y^2−x\le0$. If $V=a\pi/b$, where $a$ and $b$ are coprime positive integers, what is the value of $a+b$?
if we consider $x^2+y^2=r... | The cross-section of this volume is the intersection of the circles $(x-1/2)^2+y^2=1$ and $x^2+y^2=1-z$. Draw the picture to see that the intersection breaks into two pieces: that bounded on top by the first circle, and that by the second. The integral is
$$\int_0^1 dz \: \left [ \int_0^{1-z} dx \: \int_0^{\sqrt{1/4 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show $ \frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1 $ given $xyz = 1$ Please help me prove the equality: If $xyz=1$, prove that
$$
\frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1
$$
| Note that $$\dfrac{x}{xy+x+1} = \dfrac{xz}{xyz+xz + z} = \dfrac{xz}{xz+z+1}$$
Similarly, we get
$$\dfrac{y}{yz+y+1} = \dfrac1{z+1+\dfrac1y} = \dfrac1{z+1+xz}$$
Now add the three terms and finish it off by canceling the numerator and denominator.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/332188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
finding the real values of $x$ such that : $x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$ How to find the real values of $x$ such that : $$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
| Here's an approach leading to a "closed formula".
The two innermost square roots are defined only, if $x\in[-2,2]$, so we can write
$x=2\cos\theta$ for some $\theta\in[0,\pi]$. My solution relies on the trig identities
$$
\sqrt{\frac{1+\cos\alpha}2}=\cos\frac\alpha2
$$
and
$$
\sqrt{\frac{1-\cos\alpha}2}=\sin\frac\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
confirm which one is correct? Let $f(z)=-(x^2+y^2)^{1/2}$ and $\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$. Help to confirm which one is correct for $\Delta f$; this or $(x^2+y^2)^{3/2}-\frac{2}{(x^2+y^2)^{1/2}}$?
thankyou.
EDIT: sorry, I meant above. Just one minus sign more.
| This is an application of the chain rule.
$$\nabla^2\!f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} . $$
If $f(x,y) = \sqrt{x^2+y^2}$ then the first order partial derivatives are:
$$\begin{array}{ccc}
\frac{\partial\! f}{\partial x} &=& \frac{x}{\sqrt{x^2+y^2}} \\
\frac{\partial\! f}{\partial ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Finding polynomial given the remainders Question: Find a polynomial $f(x) \in \mathbb{Q} (x)$ of minimal degree that has both the following properties:
When $f(x)$ is divided by $(x-1)^2$, the remainder is $2x$; and when $f(x)$ is divided by $(x-2)^3$, the remainder is $3x$.
Answer provided: $f(x)=(x-2)^3 \cdot (4x-3)+... | It might be worth noticing that this can be phrased in terms of congruences
$$
\begin{cases}
f(x) \equiv 2x \pmod{(x-1)^2}\\
f(x) \equiv 3x \pmod{(x-2)^3}.
\end{cases}
$$
There is definitely a solution because of CRT, since $(x-1)^2$ and $(x-2)^3$ are clearly coprime.
The solution is of course the same the others have ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
For which $n$ is $n\sigma(n)\equiv 2 \pmod {\phi(n)}$? How to find all of $n \in \Bbb N$ such that: $$n\sigma(n)\equiv 2 \pmod {\phi(n)}$$
$\sigma(n)$ is summation of all distinct divisors of $n$
For $p$ prime we have: $p(p+1)=p^2+p\equiv 2 \pmod {p-1}$
but how to prove for composite number : $4,6,22$ is only solution.... | Note that $d \mid \phi(n)$ implies that $n\sigma(n) \equiv 2 \pmod{d}$.
If $p^2 \mid n$ for some odd prime $p$, then $p \mid \phi(n)$. Thus $0 \equiv n\sigma(n) \equiv 2 \pmod{p}$, a contradiction. Thus $p^2 \nmid n$ for any odd prime $p$.
If $8 \mid n$, then $4 \mid \phi(n)$. Thus $0 \equiv n\sigma(n) \equiv 2 \pmod{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\sum_{n=1}^{\infty}\frac{1}{n^3+3 n^2+2 n}$ Summing this series from $0$ to $\infty$, the result is $\frac{1}{4}$. I tried a lot, but I could not get this result. I think it´s wrong.
Can anybody help me?
| $$\frac{1}{n(n+1)(n+2)}=\frac{1}{n+2} \frac{1}{n(n+1)}=\frac{1}{n+2}\left( \frac{1}{n}- \frac{1}{n+1} \right)
=\frac{1}{n(n+2)}-\frac{1}{(n+1)(n+2)}=\frac{1}{2}\frac{1}{n}-\frac{1}{n+1}+\frac{1}{n+2}-\frac{1}{n+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/341650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 3
} |
Positive integer $2\times 2$ matrix with two positive integer square root matrices So suppose I have this matrix:
$$M=\begin{pmatrix}a & b \\ c & d \end{pmatrix}$$
I am sure that every entry is a positive integer. I am trying to figure out if it has 2 square roots in which every entry is also a positive integer. I took... | The Wikipedia article is not very clearly written. Let us derive the result by ourselves. Suppose
$$
M=\begin{pmatrix}A&B\\C&D\end{pmatrix}
=R^2={\underbrace{\begin{pmatrix}a&b\\c&d\end{pmatrix}}_{R}}^2
=\begin{pmatrix}a^2+bc&b(a+d)\\c(a+d)&bc+d^2\end{pmatrix}.
$$
Then $A+D+2(\det R) = (a+d)^2$ and
$$
\begin{pmatrix}A+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/342049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$ Show that
$$\begin{aligned} \int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2}
\right)dx &= \frac{7\pi^3}{108} \\ \int_0^{\pi/3}x\log^2
\left(2\sin\frac{x}{2} \right)dx &= \frac{17\pi^4}{6480}\end{aligned}$$
*
*... | The best way I see to do the simpler integral (the one without the $x$ in front) is to substitute $u=2 \sin{(x/2)}$ and expand the resulting integrand in a series. To wit, upon doing the substitution, we get
$$\int_0^{\pi/3} dx \: \log^2{[2 \sin{(x/2)}]} = \int_0^1 du \frac{\log^2{u}}{\sqrt{1-u^2/4}}$$
Note that
$$\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/342777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 3,
"answer_id": 1
} |
bernoulli equations1 Question:
Solve the following bernoulli equations.
$$xy'+y=y^{-2}$$
I solving:
$$w=y^1+2$$ $$,$$
$$\frac{w'}{1+2}+1w=1$$
$$\frac{-w'}{3}=-1$$
$$\frac{1}{3}(\frac{-1}{3}w'-w)=-1$$
$$d(\frac{-1}{3}w)=-1$$
$$\frac{-1}{3}w=-1+C$$
$$= \left\{
\begin{array}{l l}
w=9 & \quad \\\
w=y^3 & \quad \... | First observe that if $x = 0$ then $y = 1$ is the solution.
Next, assume $x \neq 0$. The equation can be written as
$$y^2y' + \frac{y^3}{x} = \frac{1}{x}.$$
Put $w := y^3$ then $w' = 3y^2y'$ and the equation becomes linear:
$$\frac{w'}{3} + \frac{w}{x} = \frac{1}{x}.$$
This can be solved by separation of variables. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/344700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is the remainder when $25^{889}$ is divided by 99? What is the remainder when $25^{889}$ is divided by 99 ?
$25^3$ divided by $99$ gives $26$ as a remainder.
$25*(25^3)$ divided by $99$ gives (remainder when $25*26$ is divided by $99$) as a remainder.
i.e. $25*(25^3)$ divided by $99$ gives $56$ as a remainder.
$(2... | $\rm mod\ 9\!:\ 25^3\equiv (-2)^3\equiv -8\equiv 1\:\Rightarrow\:n = 25^{889}\equiv 25^{889\ mod \ 3}\equiv 25\equiv \color{#0A0}{-2}$
$\rm mod\ 11\!:\ 25^5\equiv 5^{10}\equiv 1\:\Rightarrow\:n = 25^{889}\equiv 25^{889\ mod\ 5}\equiv 25^4 \equiv 3^4\equiv (-2)^2\equiv \color{#C00}4$
$\rm mod\ 9\!:\ \color{#0A0}{{-}2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Integral Solution Technique Could somebody explain the processes in the following integral solution?
$$\int_{0}^{\infty}f(x)dx, $$
$\text{ where } f(x) = \frac{x^{3}}{\theta^{2}}e^{-x^{2}/(2\theta^{2})}$
$$\int_{0}^{\infty}\frac{x^{3}}{\theta^{2}}e^{-x^{2}/(2\theta^{2})}dx \dots (1)$$
$$= \int_{0}^{\infty}x^{2}e^{-x^{2... | Doesn't answer the question, but I'd do:
$$
\int_0^\infty x^3 e^{- x^2 / 2 \theta^2} dx
$$
As Caran-d'Ache says, try to simplify the exponent. Try $y = x^2 / 2 \theta^2$,
that is $x = \theta \sqrt{2 y}$. When $x = 0$, $y = 0$; $x = \infty$ gives $y = \infty$. So :
$$
\begin{align*}
d x
&= \theta \sqrt{2} \frac{d y}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove $\frac{1}{bc + cd + da -1} + \frac{1}{ab + cd + da -1} + \frac{1}{ab + bc + da -1} + \frac{1}{ab + bc + cd -1} \le 2$ If $a,b,c,d > 0$ and $abcd = 1$, prove that the following inequality holds:
$$\frac{1}{bc + cd + da -1} + \frac{1}{ab + cd + da -1} + \frac{1}{ab + bc + da -1} + \frac{1}{ab + bc + cd -1} \le 2$... | By inequality of arithmetic and geometric means, $bc+da\ge 2\sqrt{abcd}=2$, so
$$\frac{1}{bc+cd+da-1}\le\frac{1}{1+cd}.\tag{1}$$
Similarly,
$$\frac{1}{ab+bc+da-1}\le\frac{1}{1+ab}.\tag{2}$$
$(1)+(2)$ gives
$$\frac{1}{bc+cd+da-1}+\frac{1}{ab+bc+da-1}\le\frac{1}{1+cd}+\frac{1}{1+ab}=\frac{2+ab+cd}{1+ab+cd+abcd}=1.\tag{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find minimum of $\frac{a+3c}{a+2b+c}+\frac{7a+6b+3c}{a+b+2c}+\frac{c-a}{2a+b+c}$ for non-negative reals
Let $a, b, c\ge 0$, not all zero. Find the minimum of
$$N = \frac{a+3c}{a+2b+c}+\frac{7a+6b+3c}{a+b+2c}+\frac{c-a}{2a+b+c}. $$
| Hint: Let $x,y,z$ be the denominators of the three fractions, and express $a,b,c$ in terms of $x,y,z$. Then write $N$ in terms of $x,y,z$. After some simplification we will get terms like $\frac{x}{z},\frac{x}{y},\frac{y}{x},\frac{y}{z},\cdots$Then consider inequality $\frac{x}{y}+\frac{y}{x}\geqslant2$ if $xy$ is posi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/351976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
is this relation true? i found this relation in old notes , it's mentioned with no proof , and i want to know if it's true or false
the relation says ,
$$ \sum_{k=1}^n n^6 = \frac{n^7}{7} + \frac{n^6}{2} + \frac{5}{2} n^5 - \frac{15}{2}n^4 - \frac{31}{6}n^3 + \frac{11}{21}n - 8 $$
is this relation true or not ?
and... | Use $$(r+1)^{n+1}-r^{n+1}=\sum_{0\le s\le n}\binom {n+1}s r^s$$
to find $\sum_{1\le t \le n}r^s$ for $s=1,2,3,5,6$
For $s=0, \sum_{1\le r \le n}r^0=\sum_{1\le r \le n}1=n$
For $n=2, (r+1)^2-r^2=2r+1$
Put $r=1,2,\cdots,n-1,n$ and add to get $$(n+1)^2-1^2=2\sum_{1\le s\le n}r+\sum_{1\le s\le n}1$$
$$\implies 2\sum_{0\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/352734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Ordered triples solution to system of equations How many ordered triples $(x,y,z)$ of integer solutions are there to the following system of equations?
$$
\begin{align}
x^2+y^2+z^2&=194 \\
x^2z^2+y^2z^2&=4225
\end{align}
$$
| We have $x^2 + y^2 = \dfrac{4225}{z^2}$. Plugging this into the first equation, we get that
$$z^2 + \dfrac{4225}{z^2} = 194 \implies z^2 + \left(\dfrac{65}z\right)^2 = 194$$ Once you have this you should be able to proceed and get the answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/353685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Variable change in integral I need to calculate the integral $$\int_0^{2\pi}\sqrt{1-\cos\,x}dx$$ but when I try the substitution with $u = 1-\cos\,x$ both integration limits goes to 0.
The same happens using the t-substitution.
| Hint: Note that
\begin{align}
1-\cos(x)
=
&
1-\cos\left(2\cdot \frac{x}{2}\right)
\\
=
&
1+\sin^2\left(\frac{x}{2}\right)-\cos^2\left(\frac{x}{2}\right)
\\
=
&
\sin^2\left(\frac{x}{2}\right)+\sin^2\left(\frac{x}{2}\right)
\\
=
&
2\sin^2\left(\frac{x}{2}\right)
\end{align}
implies
$$
\int_{0}^{2\pi}\sqrt[2\,]{1+\cos(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Choosing teams with minimum number of boys and girls. I need to find different number of teams I can make with 6 people that needs to have at least 2 girls and 2 boys. There are 8 girls and 12 boys. So the way I think is I need to find total number of different teams that I can make first and subtract all boys team, al... | if boys divided into such as Tom,Jack,Jerry....then
$$X={20\choose 6} - {8\choose6}-{12\choose6}-{8\choose5}{12\choose1}-{12\choose5}{8\choose1}=30800 $$
else it's a naive problem
just 3
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/356926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How find this $a_{1}+a_{2}+\cdots+a_{500}=b_{1}+b_{2}+\cdots+b_{500}$? Let $A=\{1^2,2^2,3^2,\cdots,1000^2\}$. How to prove :
There exist $A_{1}=\{a_{1},a_{2},a_{3},\cdots,a_{500}\}\subset A$, $A_{2}=\{b_{1},b_{2},\cdots,b_{500}\}\subset A$, such that $A_{1}\bigcup A_{2}=A,A_{1}\bigcap A_{2}=\varnothing$, and $a_{1}+a_... | Note that
$$(k+1)^2+(k+2)^2 = k^2+(k+3)^2\quad -4.$$
Therefore,
$$\tag1(k+1)^2+(k+2)^2+(k+4)^2+(k+7)^2 = k^2+(k+3)^2+(k+5)^2+(k+6)^2.$$
Now partition $A$ into $125$ groups of $8$ consecutive numbers each and distribute each such group among $A_1$ and $A_2$ accoring to $(1)$. This works because $1000$ is a multiple of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Strategies to prove inequalities with interval notation How to prove a inequalities with interval notation, for example:
Find minimum of $a^3+b^3+c^3$ with $a,b,c \in [-1;\infty), a^2+b^2+c^2=9$
| Clearly $a, b, c$ cannot be all negative.
If $a, b, c$ are all non-negative, then by Power mean inequality $(\frac{a^3+b^3+c^3}{3})^2 \geq (\frac{a^2+b^2+c^2}{3})^3=27$, so $a^3+b^3+c^3 \geq 9\sqrt{3}$ with equality when $a=b=c=\sqrt{3}$.
If exactly 1 of $a, b, c$ is negative, then WLOG assume that $c$ is negative, so ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to factorise $x^6 + 1$? How do I factorise this? I already said that
$$x^6 + 1 = 0 \implies (x^3)^2 + 1 = 0$$
so we then get
$$(x^3)^2 = - 1$$
and then there are no real roots. Similar thing happens if I try it as $(x^2)^3$.
How would I then go about factorising this? I know its possible as
$$(x^6 + 1) = (x^2 + 1)... | If $x^6+1=0$, then $x^3$ must be either $i$ or $-i$ (because $(x^3)^2=x^6=-1$). Hence, one must find the cube roots of $\pm i$. The numbers $i$ and $-i$ are easily seen to be cube roots of each other.
Now, $(a+bi)^3=a^3+3a^2bi+3a(bi)^2+(bi)^3=a^3+3a^2bi+3ab^2i^2+b^3i^3=a^3+3a^2bi-3ab^2-b^3i=(a^3-3ab^2)+(3a^2b-b^3)i$. I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
How to find the eigenvalues and Jordan canonical form of this matrix Question:
let $a_{i,j}\in R,A=(a_{i,j})_{n\times n} $,and
$a_{i,j}=\begin{cases}
1&i+j\in\{n,n+1\}\\
0&i+j\notin\{n,n+1\}
\end{cases}$
that's meaning:
$$A=\begin{bmatrix}
0&0&0&\cdots&0&1&1\\
0&0&0&\cdots&1&1&0\\
0&0&\cdots&1&1&0&0\\
\vdots&\vdots&\v... | Although Ewan Delanoy was more quick than I, :-) I’ll continue user1551‘s answer.
It seems the following. Denote the determinant of the $n\times n$ matrix $xI-A$ as $q_n(x)$. Then we can calculate the first polynomials:
$q_1=x-1$
$q_2=x^2-x-1$
$q_3=x^3-x^2-2x+1$
$q_4=x^4-x^3-3x^2+2x+1$
$q_5=x^5-x^4-4x^3+3x^2+3x-1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Inequality: $\frac{a^2+b^2}{(a+b)^4}+\frac{b^2+c^2}{(b+c)^4}+\frac{c^2+d^2}{(c+d)^4}+\frac{d^2+a^2}{(d+a)^4} \leq \frac{1}{abcd}$ If $ab+bc+cd+da\leq 8$ and $a,b,c,d \in \mathbb{R}_{+},$how can I prove the following inequality :
$$\frac{a^2+b^2}{(a+b)^4}+\frac{b^2+c^2}{(b+c)^4}+\frac{c^2+d^2}{(c+d)^4}+\frac{d^2+a^2}{(... | that is not a hard question !
$\frac{a^2+b^2}{(a+b)^4}\leq \frac{a^2+b^2}{16a^2b^2}$$\Longrightarrow$
$\frac{a^2+b^2}{(a+b)^4}+\frac{b^2+c^2}{(b+c)^4}+\frac{c^2+d^2}{(c+d)^4}+\frac{d^2+a^2}{(d+a)^4} $$=\frac{1}{8}$$(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{cd}+\frac{1}{ad})=\frac{1}{8}\cdot{\frac{ab+bc+cd+ad}{abcd}}$
$\leq \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find all solutions: $x^2 + 2y^2 = z^2$ I'm use to finding the solutions of linear Diophantine equations, but what are you suppose to do when you have quadratic terms? For example consider the following problem:
Find all solutions in positive integers to the following Diophantine equation
$x^2 + 2y^2 = z^2$
I'd usuall... | We can approach this in the same manner as Pythagorean Triples.
Let's only look for primitive solutions, $\gcd(x,y,z)=1$. Since
$$
z^2-x^2=2y^2
$$
$z$ and $x$ must have the same parity. That means both $z-x$ and $z+x$ are even so $y$ must be even. Therefore, for the triple to be primitive, $x$ and $z$ must be odd. Let ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 5
} |
Proving existence of a square-free sequence I found this problem and a solution sketch in a MathOverflow answer, and I thought it was nice enough to deserve more attention and a properly written solution.
Problem:
Prove that for each natural number $n$, there is some natural number $r$ for which the $n$ integers $r+... | Let $p_1, p_2, \cdots , p_k$ be all the prime numbers less than $n^2$ (for some $k$). For any $K > k$, let $p_{k+1}, p_{k+2}, \cdots , p_K$ be the next prime numbers after that (larger than $n^2$).
For $1 \le i \le k$, there is at least one value of $r$ $\pmod {p_i^2}$ that satisfies $r + 1^2, r+2^2, \cdots , r+n^2 \n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Strings and Substrings So here is one of the last homeworks we are doing in my Discrete math class. It seems like it should be simple but I am really stuck. Any help would be greatly appreciated.
*
*Find the ordinary generating series with respect to length
for the strings in $\{0,1,2\}^*$ having no "$22$" su... | You can uniquely generate the set of strings using $(0|1|20|21)^∗(2|ϵ)$, hence your generating function is $\frac{1}{1 - (x + x + x^2 + x^2)} \cdot (1 + x) = \frac{1 + x}{1 - 2(x + x^2)}$.
Let $\frac{1 + x}{1 - 2(x + x^2)} = a_0 + a_1x + a_2x^2 + \cdots$. Then by multiplying both sides by $1 - 2(x + x^2)$ and looking a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
expand $ \arctan\left(\frac{3x+2}{3x-2}\right)$ into pwer series, find radius of convergence (check solution) I would be grateful if someone could check what I've worked out:
$$ f(x)=\arctan\left(\frac{3x+2}{3x-2}\right)\implies f'(x)=\frac{1}{1+(\frac{3x+2}{3x-2})^2}\cdot \frac{3(3x-2)-3(3x+2)}{(3x-2)^2}$$
$$=\frac{(... | Here is what you are missing,
$$ f(x)=\int f'(x)dx=-\frac{3}{2} \sum_{k=0}^{\infty}\left((-1)^k\frac{3}{2} \frac{x^{2k+1}}{2k+1}\right) +C $$
$$ \implies f(0) = \arctan(-1) = 0 + C \implies C=-\frac{\pi}{4}. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/370087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Distributing identical objects to identical boxes We have 6 identical things to be distributed in 4 identical boxes such that empty boxes are allowed the find the number of ways to distribute the things ?
| The result is the number of partitions of 6 into a sum of 4 integral summands $\geq 0$.
The possibilities are
$$
6 = 6 + 0 + 0 + 0 \\
= 5 + 1 + 0 + 0 \\
= 4 + 2 + 0 + 0 \\
= 4 + 1 + 1 + 0 \\
= 3 + 3 + 0 + 0 \\
= 3 + 2 + 1 + 0 \\
= 3 + 1 + 1 + 1 \\
= 2 + 2 + 2 + 0 \\
= 2 + 2 + 1 + 1
$$
so the answer is $9$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/373202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Sum of the first $n$ triangular numbers - induction Question:
Prove by mathematical induction that $$(1)+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)=\frac{1}{6}n(n+1)(n+2)$$ is true for all positive integers n.
Attempt:
I did the the induction steps and I got up to here:
$$RTP:\frac{1}{6}n(n+1)(n+2)+(1+2+3+\cdots+n+(n+1))=\f... | $$\frac{1}{6}n(n+1)(n+2)+(1+2+3+\cdots+n+1)=$$
$$=\frac{1}{6}n(n+1)(n+2)+\frac{(n+1)(n+2)}{2}=$$
$$=\frac{1}{6}n(n+1)(n+2)+\frac{3(n+1)(n+2)}{6}=$$
$$=\frac{1}{6}(n+1)(n+2)(n+3)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/376284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Continued fraction for $\sqrt{14}$ I'm referencing this page: An Introduction to the Continued Fraction, where they explain the algebraic method of solving the square root of $14$.
$$\sqrt{14} = 3 + \frac1x$$
So, $x_0 = 3$, Solving for $x$, we get
$$x = \frac{\sqrt{14} + 3}5$$
However, in the next step, how do we get t... | $\sqrt{14}=3+\sqrt{14}-3=3+\frac{1}{\frac{\sqrt{14}+3}{5}}\implies x_0 = 3$
$\frac{\sqrt{14}+3}{5}=\frac{6+\sqrt{14}-3}{5}=1+\frac{\sqrt{14}-2}{5}=1+\frac{1}{\frac{\sqrt{14}+2}{2}} \implies x_1 = 1$
$\frac{\sqrt{14}+2}{2}=\frac{5+\sqrt{14}-3}{2}=2+\frac{\sqrt{14}-2}{2}=2+\frac{1}{\frac{\sqrt{14}+2}{5}} \implies x_2 = 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How do I factorize equations of the form $x^2 + Bxy + Cy^2 = 0$ Given equation
$$ x^2 + Bxy + Cy^2 = 0. $$
I want to factorize it in the form
$$ (x + my)(x + ny) = 0. $$
What are the values of $m$ and $n$ in terms of $B$ and $C$?
I tried writing the equation as
$$ x^2 + Bxy + Cy^2 = (x + kCy)(x + \dfrac{1}{k}y) = 0. $... | Nice solution! Please note that Glen O has answered your question very well, and what comes below does not. Its purpose is to give a slightly different view of the matter, and to connect it with other things that you know.
Let us factor $4x^2+4Bxy+4Cy^2$. Completing the square, we get $(2x+By)^2-(B^2-4C)y^2$. If $B^2-4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Closed form for $\sum_{n=-\infty}^{\infty} \frac{1}{n^4+a^4}$ I've been working with series this week and came across a couple that have been bugging me.
I'm looking for the closed form of:
$$
J(a)=\sum_{-\infty}^{\infty} \frac {1} {n^4+a^4}
$$
As with the other one I posted, solving for the closed form seems like a fu... | Rather "magical" approach :
Recall the Mittag-Leffler meromorphic expansion of the $\cot$ ($\coth$, resp.) function :
$$\sum_{n=-\infty}^{\infty}\frac{1}{n^2-a^2} = -\frac{\pi\cot\pi a}{a} $$
holding for any complex $a$ (exept the poles obviously) - this fact we will use later as change $a\rightarrow a i$.
From this we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 1
} |
Examine the continuity of $f(x)=x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{(1+x^2)^2}+...+ \frac{x^2}{(1+x^2)^n}+....$ at $x=0$
Examine the continuity of $$f(x)=x^2+\frac{x^2}{(1+x^2)}+\frac{x^2}{(1+x^2)^2}+...+ \frac{x^2}{(1+x^2)^n}+....$$ at $x=0$
I tried to solve the problem using $$\lim_{x \to0^+}f(x)=\lim_{x \to0^-}f(x)... | Note that it's a geometric series, as Haral Hanche-Olsen commented. You can write $f(x)$ as
$$f(x) = x^2 \frac{1-\frac{1}{(1+x^2)^{n+1}}}{1-\frac{1}{1+x^2}}=x^2\frac{(1+x^2)^{n+1}-1}{x^2(1+x^2)^{n}} = \frac{(1+x^2)^{n+1}-1}{(1+x^2)^{n}}$$
This function is continuous at $x=0$ in my sense...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/389260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Laurent Series Expansion Give the Laurent series development of the function $f(z)=\frac{1}{z(z-1)(z-2)}$ in the three rings $A_1=\{z:|z|<1\}$, $A_2=\{z:1<|z|<2\}$ and $A_3=\{z:2<|z|\}$.
I have gotten the partial fraction form $f(z)=\frac{1}{2z}-\frac{1}{z-1}+\frac{1}{z-2}$
For $A_1$, I think it's the standard Taylor e... | First of all, the partial fraction decomposition is
$$
f(z) = \frac{1}{2z} - \frac{1}{z - 1} + \frac{1}{2(z - 2)}.
$$
The Laurent series converges on an annulus, so $A_1$ must not contain the origin:
$$
A_1 = \left\{z : 0 < |z| < 1 \right\}.
$$
Then, you can just use the usual Taylor expansion around $z = 0$:
$$
\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
conic sections, ellipse A particle is travelling clockwise on the elliptical orbit given by
$$\displaystyle \frac{x^2}{100} + \frac{y^2}{25} = 1$$
The particle
leaves the orbit at the point $(-8, 3)$ and travels in a straight line tangent to the ellipse.
At which point will the particle cross the $y$-axis?
| I like to work with homogeneous coordinates.
The ellipse is represented by
$$C = \begin{bmatrix} \frac{1}{100} & 0 & 0 \\
0 & \frac{1}{25} & 0 \\
0 & 0 & -1 \end{bmatrix}$$
the 3×3 matrix such that
$$ \left. \begin{pmatrix} x & y & 1 \end{pmatrix} \begin{bmatrix} \frac{1}{100} & 0 & 0 \\
0 & \frac{1}{25} & 0 \\
0 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Solve $\sqrt[3]{x+10}-\sqrt[3]{x-10}=2$.
Solve $\sqrt[3]{x+10}-\sqrt[3]{x-10}=2$.
I tried cubing the both sides but things then go very ugly. Are there simpler way to solve it? Thanks.
p.s. The answers are $\pm 6\sqrt 3$.
| $\displaystyle\sqrt[3]{x+10}-\sqrt[3]{x-10}=2---(1)$
Let $y=x+10$
$(1)$ becomes
$\displaystyle\sqrt[3]{y}-\sqrt[3]{y-20}=2$
$\displaystyle\sqrt[3]{y}=2+\sqrt[3]{y-20}$
Cube both sides to get
$\displaystyle{y}=8+12 (y-20)^{\frac{1}{3}}+6(y-20)^{\frac{2}{3}}+y-20$
$\displaystyle{0}=-12+ 6 (y-20)^{\frac{1}{3}}(2+(y-20)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving two equations involving the greatest common divisor
Show or prove that $$\gcd \left(\frac{a^{2m}-1}{a+1} ,a + 1\right )=\gcd(a + 1 , 2m),$$
and that
$$\gcd \left(\frac{a^{2m + 1}+1}{a+1} , a + 1\right)=\gcd(a + 1 , 2m + 1).$$
| Let integer $d\ne0$ divides $a+1$ i.e., $a+1=c\cdot d$(say) where $c\ne0$ is some integer
$\implies a=c\cdot d-1$
$$\gcd \left(\frac{a^{2m}-1}{a+1} ,a + 1\right )$$
$$=\gcd \left(\frac{(c\cdot d-1)^{2m}-1}{c\cdot d} ,c\cdot d\right )$$
$$=\gcd \left( (c\cdot d)^{2m-1}-\binom{2m}1(c\cdot d)^{2m-2}+ \binom{2m}2(c\cdot d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
On Sixth Powers $x_1^6+x_2^6+\dots+x_6^6 = z^6$ Fourteen years ago, in 1999 (has it been that long?) Merignac started a search for,
$$x_1^6+x_2^6+\dots+x_6^6 = \color{red}z^6$$
using the five congruence classes,
$$\begin{aligned}
&b^6(x_1^6+x_2^6+\dots+x_5^6)+(1x_6)^6+(42x_7)^6 = z^6\\
&b^6(x_1^6+x_2^6+\dots+x_5^6)+(2x... | The restriction to $b = 42$ is correct because any solution of the Diophantine equation
$$x_1^6+x_2^6+\dots+x_6^6 = z^6$$
would have to meet constraints associated with each of the prime factors of 42:
a) If $a \not\equiv 0 \pmod{2}$ then $a^6 \equiv 1 \pmod{8}$;
b) If $a \not\equiv 0 \pmod{3}$ then $a^6 \equiv 1 \pmod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Floor of log equation $S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - \frac{10^{\lfloor\log_{10}(x)\rfloor+1}-10}{9}$ I must find 'x' and I don't know how to solve the following equation.
Does it have a solution? How can I solve it?
$$
S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - \frac{10^{\lfloor\log_{10}(x)\rfloor+... | For an integer $k\ge 0$, if $10^k \le x < 10^{k+1}$, then $\lfloor \log_{10}x\rfloor = k$ so
$$
S(x) = (k+1)x -\frac{10^{k+1}-10}{9}=(k+1)x-\frac{10}{9}(10^k-1)
$$
So, for example, if $1\le x \le 9$, then $S(x) = x$. If $10\le x \le 99$, then $S=2x-10$. If $100\le x\le 999$, then $S=3x-110$, and so on.
The first few $S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/396339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What am I doing wrong in these quartic formula calculations? I was a bit surprised that there is a general formula for the roots of a quartic equation, so I decided to test Wikipedia's version of it myself. To my surprise, I have arrived at a correct answer only once in about five attempts, using only integer coefficie... | My error was made in rationalizing the denominator of $v$. I multiplied both parts of the fraction by $Q^2$, but rendered the fraction as
$$\frac{35 + 18i\sqrt{3} + 13\left(\sqrt[3]{35 + 18i\sqrt{3}}\right)^2}{3(35 + 18i\sqrt{3})}$$
(The first $35 + 18i\sqrt{3}$ should be $(35 + 18i\sqrt{3})\sqrt[3]{35 + 18i\sqrt{3}}$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/396489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Solve for $x$, $3\sqrt{x+13} = x+9$ Solve equation: $3\sqrt{x+13} = x+9$
I squared both sides and got $9 + x + 13 = x^2 + 18x + 81$
I then combined like terms $x^2 + 17x + 59 = 0$
I then used the quadratic equation $x= -\frac{17}2 \pm \sqrt{\left(-\frac{17}2\right)^2-59}$
However, the answer is 3
| When you square the LHS, it should be $9(x+13)$ not $9+x+13$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/398051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
System of two simple modular equations $$x \equiv -7 \mod 13$$
$$x \equiv 39 \mod 15$$
I need to find the smallest x for which these equations can be solved. I've been always doing this using Chinese Reminder Theorem, but it seems that it doesn't work here, I'm not sure why though. Let's see:
$N = 13*15 = 195$
$N_{1} =... | The way I usually go about these is to solve the two systems
$$
\mbox{$\begin{align}
x_{13}&\equiv1\pmod{13}\\
x_{13}&\equiv0\pmod{15}
\end{align}$}
\qquad\text{and}\qquad
\begin{align}
x_{15}&\equiv0\pmod{13}\\
x_{15}&\equiv1\pmod{15}
\end{align}
$$
Then combine these to get the solution to the equation given. These c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/398327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Rearranging a series. Consider the following two series.
$$\sum a_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+ \cdots$$ $$\sum b_n = 1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+ \cdots$$
I was able to say that $a_n = (-1)^{n+1} \frac{1}{n}$ while $b_n = \frac{1}{4n-3}+\frac{1}{4n-1}-\frac{1}... | Since
$$
\small
b_n=\frac{8n-3}{2n(4n-1)(2n-3)}\sim\frac{1}{2n^2}
$$
The series $\sum b_n$ absolutely converges. To find its sum we need to work a little. Note that
$$
\small
\begin{align}
\sum\limits_{k=1}^n b_k
&=\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots-\frac{1}{4n}\right)+\color{green}{\left(\frac{1}{2n}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/405542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve system of first order differential equations I have to solve differential systems like this:
$$
\left\{
\begin{array}{c}
x' = 3x - y + z \\
y' = x + 5y - z \\
z' = x - y + 3z
\end{array}
\right.
$$
Until now I computed the eigenvalues $k = \{2,4,5\}$ by solving the equation resulted from this determinant of thi... | To find the eigenvalues of: $$A =\begin{bmatrix}3 & -1 & 1\\1 & 5 & -1\\1 & -1 &3\end{bmatrix},$$
we set up $|A - \lambda I| = 0$ and solve the characteristic polynomial, so we have:
$$|A -\lambda I| = \begin{bmatrix} 3-\lambda & -1 & 1\\1 & 5-\lambda & -1\\1 & -1 &3 -\lambda \end{bmatrix} = 0$$
From this, we get t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/406055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Convergence/Divergence of a particular infinite nested radical Is it known if the following infinite nested radical converges or diverges (for $n \in \mathbb N$)?:
$$R(n) = \sqrt{n+\sqrt{(n+1)+\sqrt{(n+2)+ \cdots}}}$$
I recently became interested in these problems when reading about Ramanujan's (famous?) formula: $$\sq... | If you accept
$$
\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{\cdots}}}}=\sqrt{1+\phi}=\frac{1+\sqrt{5}}{2}
$$
then
$$
\begin{align}
\phi < R(1) &< \sqrt{1+\sqrt{2+\sqrt{2^2+\sqrt{2^4+\sqrt{2^8+\cdots}}}}}\\
&=\sqrt{1+\sqrt{2}\phi}
\end{align}
$$
so
$1.615<R(1)<1.815$ and as you say you can get the other $R(n)$ from there.
Edit:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/410413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Find volume using double integrals? Question: Use double integral to find the volume of the solid enclosed by the spheres $x^2+y^2+z^2=1$ and $x^2+y^2+(z-1)^2=1$
Alright so I tried to doing this by myself and I'm not sure if this is right. Could someone check over my work?
Curve of intersection:
\begin{align*}
x^2 + y^... | Besides to @Shuhao's approach you can use the Spherical coordinates as well. In fact the limits in this system would be:
$$r|_{\frac{1}{2\cos(\theta)}}^1,~~\theta|_0^{2\pi},~~\phi|_0^{\pi/3}$$
Note that two creatures intersects at $z=1/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/411261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Find the largest number having this property. The $13$-digit number $1200549600848$ has the property that for any $1 \le n \le 13$, the number formed by the first $n$ digits of $1200549600848$ is divisible by $n$ (e.g. 1|2, 2|12, 3|120, 4|1200, 5|12005, ..., 13|1200549600848 using divisor notation).
Question 1: Find th... | Let $n_k = 10^{k - 1} a_1 + 10^{k - 2} a_2 + \cdots + a_k$ for positive integer $k$. Also,
$$\begin {eqnarray}
n_{k + 1} & = & 10^k a_1 + 10^{k - 1} a_2 + \cdots + 10 a_k + a_{k + 1} \nonumber \\ & = & 10 n_k + a_{k + 1}. \tag{1}
\end {eqnarray}$$
Since $k | n_k$, we have, by $(1)$, that
$$\begin {equation*}
\begin {ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/411897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Calculating $\sum^{10}_{k=1}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right)$
Find the value of $$\sum^{10}_{k=1}\left (\sin\left (\frac{2k\pi}{11} \right )+i\cos\left (\frac{2k\pi}{11}\right ) \right)$$
My approach:
Since $\cos\theta + i\sin\theta = e^{i\theta}$, we can write the given equation as:
$$\begin... | You went wrong when you turned the sum into a product (which you'd have to do in order to sum the indices like that).
Instead, multiply (i) by $1=e^{i2\pi}$ and reindex to get $$i\sum_{k=1}^{10}e^{i2\pi\left(1-\frac{k}{11}\right)}=i\sum_{k=1}^{10}e^{\frac{i2\pi}{11}},$$ then substitute $\omega=e^{\frac{i2\pi}{11}}$ to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong?
$$(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$$
Thanks
| We know $$(x+y+z)^3=x^3+y^3+z^3+3(x+y)(y+z)(z+x)$$
If $x+y+z=0, x^3+y^3+z^3=-3(x+y)(y+z)(z+x)=-3(-z)(-x)(-y)=3xyz$
Alternatively if $x+y+z=0,$
$x^3+y^3+z^3=(x+y)^3-3xy(x+y)+z^3=(-z)^3-3xy(-z)+z^3=3xyz$
Put $x=a-b,y=b-c,z=c-a$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 12,
"answer_id": 3
} |
Volume bounded by sphere $x^2+y^2+z^2=a^2$ and cylinder $x^2+y^2=a|x|$ What is the volume bounded by the sphere $x^2+y^2+z^2=a^2$ and the cylinder $x^2+y^2=a|x|$?
The answer can be in terms of the value $a$ (or $r$). Does someone know how to do this?
Thank you in advance!
| The figure is symmetric, with equal volume in each of the eight octants, so we focus on the first octant, and multiply by $8$. Let's work in cylindrical coordinates. In that case, our function is $z=\sqrt{a^2-r^2}$, and our region of integration is bounded by
$$0\le\theta\le\frac{\pi}{2}\\0\le r\le a\cos\theta$$
The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to integrate $\frac{x+2}{x^2+2x+2}$ with the substitution method This is what I got so far:
$$\frac{x+2}{x^2+2x+2} = \frac{x+2}{(x+1)^2+1} = \int \frac{x}{(x+1)^2+1} + 2 \times \int \frac{1}{(x+1)^2+1}$$
I know the last integral $= \arctan(x+1) + c$, where $c$ is a constant, but I don't know how to integrate the fi... | HINT: You want one of the integrals to integrate to give a logarithmic function. Take this into account when you split up the fractions and you should be fine.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/415430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The system of equations $x^2 + y^2 - x - 2y = 0$ and $x + 2y = c$ I have
$(1.) \quad x^2 + y^2 - x - 2y = 0 \\
(2.) \quad x + 2y = c$
Solving for $y$ in $(2.)$ gives
$y = (c - x) / 2$
Is there a way to simplify equation $(1.)$?
Because at the end I arrive at
$c^2 - 2x - c = 0$
and can't proceed. Need to get typical fo... | You're intersecting a circle with the straight lines parallel to $x+2y=0$.
The circle can be written, by completing the squares, as
$$
x^2 -2\frac{1}{2} x + \frac{1}{4} + y^2 - 2y + 1 = \frac{1}{4}+1
$$
or, writing the squares,
$$
\biggl(x-\frac{1}{2}\biggr)^2+(y-1)^2=\frac{5}{4}
$$
Thus the lines you're looking for mu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How find $a$ such that $x^2-\sqrt{a-x}=a$ has exactly two real solutions Consider the equation
$$ x^2-\sqrt{a-x}=a.$$ I wish to determine the values of $a$ for which the above equation has exactly two real solutions (for $x$).
My idea:
$$a-x=(x^2-a)^2=x^4-2ax^2+a^2\Longrightarrow f(x)=x^4-2ax^2+x+a^2-a=0$$ and we must ... | Continuing from the moment of squaring both parts of the equation, we will solve the equation with respect to the parameter $a$. From $x^4-2ax^2+x-a+a^2=0$ we have:
$$a^2-\left(2x^2+1\right)a+\left(x^4+x\right)=0 \Rightarrow \left(a-x^2-x\right)(a-x^2+x-1)=0.$$
Now find $x$ in terms of $a$:
$$x=\frac{1\pm \sqrt{4a-3}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
How to determine the rank and determinant of $A$? let $A$ be $$A_{a} = \begin{pmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1\\ 1 & 1 & a & 1\\ 1 & 1 & 1 & a \end{pmatrix}$$
How can I calculate the rank of $A$ by the Gauss' methode and $\det A$?
| We know that this matrix is equivalent to $$B=\begin{pmatrix}a&0&0&1-a\\1&a-1&0&0\\1&1-a&a-1&0\\1&0&1-a&a-1\end{pmatrix}$$
and $$\det(B)=a(a-1)^3-(1-a)((1-a)^2-(a-1)((1-a)-(a-1))=(a-3)(a-1)^3$$ and $\text{rank}(A)=4$ for all values of $a$ except $1$ and $3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/417514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
proving $\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+\frac{1}{5\cdot 6}+\cdots+$ Without Induction i proved that:
$$
\begin{align}
& {} \quad \frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+\frac{1}{5\cdot 6}+\cdots+\frac{1}{(2n-1)\cdot 2n} \\[10pt]
& =\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\text{ for }n\in \math... | We have
$$\frac{1}{2k(2k-1)}=\frac{1}{2k-1}-\frac{1}{2k}=\frac{1}{2k-1}+\frac{1}{2k}-\frac{1}{k}$$
so
$$\sum_{k=1}^n\frac{1}{2k(2k-1)}=\sum_{k=1}^n\left(\frac{1}{2k-1}+\frac{1}{2k}\right)-\sum_{k=1}^n\frac{1}{k}=\sum_{k=1}^{2n}\frac{1}{k}-\sum_{k=1}^n\frac{1}{k}=\sum_{k=n+1}^{2n}\frac{1}{k}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/417626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Evaluating $\lim_{x\to1}\frac{\sqrt{x^2+3}-2}{x^2-1}$? I tried to calculate, but couldn't get out of this:
$$\lim_{x\to1}\frac{x^2+5}{x^2 (\sqrt{x^2 +3}+2)-\sqrt{x^2 +3}}$$
then multiply by the conjugate.
$$\lim_{x\to1}\frac{\sqrt{x^2 +3}-2}{x^2 -1}$$
Thanks!
| Let $t=x-1$ so $x=t+1$ and since $(1+y)^\frac{1}{2}\sim_0 1+\frac{y}{2}$ and $y^2=_0o(y)$ then we find
$$\lim_{x\to 1}\frac{\sqrt{x^2+3}-2}{x^2-1}=\lim_{t\to 0}\frac{\sqrt{t^2+2t+4}-2}{t^2+2t}=\lim_{t\to 0}2\frac{\sqrt{\frac{t^2+2t}{4}+1}-1}{t^2+2t}=\lim_{t\to 0}2\frac{t/4}{2t}=\frac{1}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/418748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Find all $n$ such that $\sigma(n)=12$ Let $ \sigma (n) = \sum_{k|n}^{}{k} $. I need to solve $\sigma(n)=12$.
Probably the following might be of use: if $n={p_1}^{a_1}{p_2}^{a_2}...{p_s}^{a_s}$ then $\sigma(n)=\frac{{p_1}^{a_1+1}-1}{p_1-1}\frac{{p_2}^{a_2+1}-1}{p_2-1}...\frac{{p_s}^{a_s+1}-1}{p_s-1}$.
| Note that
$$
\begin{align}
\sigma (1) &= 1\\
\sigma (2) &= 1+2 = 3\\
\sigma (3) &= 1+3 = 4\\
\sigma (4) &= 1+2+4 = 7\\
\sigma (5) &= 1+5 = 6\\
\sigma (6) &= 1+2+3+6 = 12\\
\sigma (7) &= 1+7 = 8\\
\sigma (8) &= 1+2+4+8 = 15\\
\sigma (9) &= 1+3+9 = 13\\
\sigma (10) &= 1+2+5+10 = 18\\
\sigma (11) &= 12
\end{align}
$$
so t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/419585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Probability and arithmetic sequence question Three numbers are chosen at random from $1,2,...2n$ with $n > 1$. Show that the probability that the numbers are in
A.P. is $\frac{3}{4n-2}$
I don't know how to deal with this. Please help.
| The total number of $3$-number combinations from $1$ to $2n$ is $\binom{2n}{3}$. Now consider $1$ as the first taken number. Then, the $3$-number A.P. with the greatest common difference will occur if the then highest chosen number is $2n-1$ as common difference is $\dfrac{(2n-1)-(1)}{2}=n-1$. Hence the combination of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/419640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Distributing two distinct objects to identical boxes Hiii,
I've been struck with a problem which deals with the distribution of two distinct objects such that p of one type and q of other type into three identical boxes.
As if it were only one object with q copies i'd have used integer partitioning, and if all objects ... | For the sake of completeness I am posting an answer using the Polya Enumeration Theorem. As was pointed out, the group acting on the boxes is $S_3$ with cycle index
$$Z(S_3) = \frac{1}{6} a_1^3 + \frac{1}{2} a_1 a_2 + \frac{1}{3} a_3.$$
The pattern repertoire is given by
$$ \frac{1}{1-X}\frac{1}{1-Y}.$$
Substituting in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/420614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $? I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a half-circle part ($\alpha_r$) and a path connecting the ... | $$\underbrace{\dfrac{x^2 \log(x)}{1+x^4} dx \to \dfrac1{x^2} \dfrac{\log(1/x)}{1+1/x^4} \dfrac{-dx}{x^2}}_{x \to 1/x}$$
Hence,
\begin{align}
I = \int_0^{\infty} \dfrac{x^2 \log(x)}{1+x^4} dx & = \int_0^1 \dfrac{x^2 \log(x)}{1+x^4} dx + \int_1^{\infty} \dfrac{x^2\log(x)}{1+x^4}dx\\
& = \int_0^1 \dfrac{x^2 \log(x)}{1+x^4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 7,
"answer_id": 0
} |
How to determine whether this series convergent or divergent? Does
$$
\dfrac{7}{19}+\dfrac{7}{19}\sqrt{\dfrac{7}{19}}+\dfrac{7}{19}\sqrt{\dfrac{7}{19}}\sqrt[3]{\dfrac{7}{19}}+\cdots+\dfrac{7}{19}\sqrt{\dfrac{7}{19}}\sqrt[3]{\dfrac{7}{19}}\cdots\sqrt[n]{\dfrac{7}{19}}+\cdots
$$
converge or diverge?
The following is my ... | It can be used Raabe's test.
We have series
$$\sum_{n=1}^{\infty} a_n, \qquad \mathrm{where } \quad
a_n = \left(\dfrac{7}{19}\right)^{1+\frac{1}{2}+\cdots+\frac{1}{n}}.
$$
We wil construct value $R_n= n \left( \dfrac{a_{n+1}}{a_n}-1\right)$. Denote
$R=\lim\limits_{n\to\infty} R_n$.
If $R<-1$, then series converges.
If... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$ I don't know if I apply for this case sin (a-b), or if it is the case of another type of resolution, someone with some idea without using derivation or L'Hôpital's rule? Thank you.
$$\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x... | Perhaps not an elegant proof but I considered this.
Use a Taylor series for $\sin$.
$\sin(x)=x+ a_1x^3 + a_2x^5 + ...$
Let $A$ be the value of the limit.
Then we get
$A=\dfrac{x^2+\frac{1}{x}-\frac{1}{x}+a_1(x^2+\frac{1}{x})^3-a_1(\frac{1}{x})^3+\ldots}{x} = \dfrac{x^2+a_1(x^2+\frac{1}{x})^3-a_1(\frac{1}{x})^3)+\ldots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
intersection of two subsets of vectors and its geometrical meaning E and F is a subset of $ \mathbb R^3$.
How to compute $ P = E \cap F $ ?
What is the geometrical meaning of E, F and P ?
$ E =\left( \begin{array}{ccc}
1 \\
2 \\
-3 \end{array} \right)+\alpha\left( \begin{array}{ccc}
1 \\
0 \\
1 \end{array} \ri... | To find the intersection, the analytic way would be to write out both planes as equations of the form $ax + by + cz + d = 0$, equate the left hand sides of each plane's equations with each other, and solve for x, y, and z.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/424004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$ If a,b,c are real numbers satisfying $a+b+c =0; a^2+b^2+c^2=1$.
Prove that $a^2 b^2 c^2 \le \frac{1}{54}$.
| There are three real numbers, and only two sign(negative, non-negative),So there are two real numbers with same sign,then without loss generality, Assume ab $\ge$ 0
$1 = a^2+b^2+c^2 = a^2+b^2+(-a-b)^2 = (a^2+b^2+ab) * 2 ≥ 3*ab * 2$
So
$ab ≤ \dfrac{1}{6}$
$a^2 b^2 c^2 = a^2b^2(a^2+b^2+2ab)=a^2b^2(\dfrac{1}{2}+ab)≤\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 0
} |
Limit involving power tower: $\lim\limits_{n\to\infty} \frac{n+1}n^{\frac n{n-1}^\cdots}$ What is the value of the following limit?
$$\large \lim_{n \to \infty} \left(\frac{n+1}{n}\right)^{\frac{n}{n-1}^{\frac{n-1}{n-2}^{...}}}$$ In general what do limits of infinite decreasing numbers strung together in familiar ways ... | For $n\ge 1$, let $$a_n=\left(\frac{n+1}{n}\right)^{\frac{n}{n-1}^{\frac{n-1}{n-2}^{...}}}$$ where the tower stops when we reach $2/1=2$, so $a_1=2$ and $$a_{n+1}=\left(\frac{n+2}{n+1}\right)^{a_n}.$$
Note that each $a_n$ is (strictly) larger than $1$, and that the sequence is decreasing from $n=2$ on: First, $a_2=9/4>... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
} |
Orthogonal basis for this indefinite symmetric bilinear form Find an orthogonal basis for the bilinear form over $\mathbb{R}$ given by $(\mathbf{x}, \mathbf{y})\mapsto\mathbf{x}^{t}A\mathbf{y}$ where $A=\begin{bmatrix} 1 & 4 & 4\\ 4 & 4 & 10\\ 4 & 10 & 16 \end{bmatrix}$.
I'm not sure if this is as easy as using Gram-S... | PREFACE: this is called Hermite reduction. It is, for example, Theorem 23 in The Arithmetic Theory of Quadratic Forms by Burton W. Jones, pages 56-59 primarily. On page 58 you clearly see an upper triangular matrix as the change of variables. The W stands for Wadsworth.
I get
$$ x^2 + 4 y^2 + 16 z^2 + 20 y z + 8 z x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A question on differentiation : Let $$f(x)=\sin^{-1}(2x\sqrt{1-x^2})$$
I found out $f'(x)$ in three methods and got three different answers !
1) Putting $x=\cos\theta$, we get $f(x)=2\cos^{-1}x$, on differentiating this we get
$$f'(x)=\frac{-2}{\sqrt{1-x^2}}$$
2) Putting $x=\sin\theta$, we get $f(x)=2\sin^{-1}x$, on ... | HINT:
As the principal value of $\sin^{-1}x$ lies $\in [-\frac\pi2, \frac\pi2]; \pi\le 2\sin^{-1}x\le \pi$
So, $$\sin^{-1}(2x\sqrt{1-x^2})=\begin{cases}
2\sin^{-1}x &\mbox{if } -\frac\pi4\le \sin^{-1}x\le \frac\pi4 \\
-\pi-2\sin^{-1}x & \mbox{if } -\frac\pi2\le \sin^{-1}x< -\frac\pi4 \\
\pi-2\sin^{-1}x & \mbox{if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/428476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Trigonometric identity: $\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$ I have to prove the following result :
$$\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$$
I tried converting $\tan\theta$ & $\cot\theta$ into $\cos\the... | You are on the right track.
writing $\tan\theta$ as$ \dfrac {\sin\theta}{\cos\theta}$ and $\cot\theta$ as $ \dfrac {\cos\theta}{\sin\theta} $, we get
$ \dfrac {\frac {\sin\theta}{\cos\theta} }{1-\frac {\cos\theta}{\sin\theta} }+\dfrac {\frac {\cos\theta}{\sin\theta} }{1-\frac {\sin\theta}{\cos\theta} }$
$= \dfrac {\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/429047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$ show that
$$\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$$
and I found
$$LHs-RHS=0.017\cdots$$
I have post this interesting problem Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$
can someone suggest any other nice method? Thank you everyone.
| A solution from a greek math forum.
$\frac{lne^{2}-ln7}{e^{2}-7} < \frac{1}{7} \Leftrightarrow \frac{1}{ln7} < \frac{7}{21-e^{2}}<\frac{7}{13,6}$
$\frac{ln8-lne^{2}}{8-e^{2}} >\frac{1}{8} \Leftrightarrow \frac{1}{ln8} < \frac{8}{24-e^{2}}<\frac{8}{16,6}$
So,
$ \frac{1}{ln7}+ \frac{1}{ln8} < \frac{7}{13,6}+ \frac{8}{16,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 3,
"answer_id": 2
} |
Evaluating $\int{{x^2 -1}\over{x^3 \sqrt{2x^4-2x^2+1}}} \mathrm dx$ How to evaluate:
$$\int{{x^2 -1}\over{x^3 \sqrt{2x^4-2x^2+1}}} \mathrm dx$$
| HINT:
First put $x^2=y$ in
$$\int{{x^2 -1}\over{x^3 \sqrt{2x^4-2x^2+1}}} \mathrm dx=\int{{x^2 -1}\over{2x^4 \sqrt{2x^4-2x^2+1}}} \mathrm 2xdx$$
to get $$\int \frac{y-1}{2y^2\sqrt{2y^2-2y+1}}dy$$
Now $2y^2-2y+1=\frac{4y^2-4y+2}2=\frac{(2y-1)^2+1}2$
Put $2y-1=\tan\theta$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/431550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
If $a,b$ are roots for $x^2+3x+1=0$.Calculating $(\frac{a}{b+1})^2 +(\frac{b}{a+1})^2$ If $a,b$ are roots for the equation $x^2+3x+1=0$.How to calculate $$\left(\frac{a}{b+1}\right)^2 +\left(\frac{b}{a+1}\right)^2$$
| We note that the expression given is a symmetric rational function of the roots of the original polynomial, so can be expressed in terms of the coefficients of the original polynomial. We then use the basic facts we know about the sum and product of the roots, and the fact that both $a$ and $b$ satisfy the polynomial t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 0
} |
Find $S$ where $S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$, why am I getting an imaginary number? $\large S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$
Multiplying by conjugate:
$\large S=\dfrac {-3}{\sqrt[3] {5+2 \sqrt {13}}-\sqrt[3]{5-2 \sqrt {13}}}$
From the original:
$\large S-2\sqrt[3]{5-2 \sqr... | You are not right: the conjugate to $S$ is
$$
\left[\sqrt[3] {5+2 \sqrt {13}}\right]^2-\sqrt[3]{5+2 \sqrt {13}}\sqrt[3] {5-2 \sqrt {13}}+\left[\sqrt[3]{5-2 \sqrt {13}}\right]^2.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/431671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Proving the inequality $\frac{a^3}{b^2-bc+c^2}+\frac{b^3}{a^2-ac+c^2}+\frac{c^3}{a^2-ab+b^2}\geq a+b+c$ I am trying to prove the following inequality
For all positive numbers $a$, $b$ and $c$ we have
$$\dfrac{a^3}{b^2-bc+c^2}+\dfrac{b^3}{a^2-ac+c^2}+\dfrac{c^3}{a^2-ab+b^2}\geq a+b+c$$
I can probably solve this by reduc... | $\dfrac{a^3}{b^2-bc+c^2}+\dfrac{b^3}{a^2-ac+c^2}+\dfrac{c^3}{a^2-ab+b^2}\geq a+b+c \iff \dfrac{a^3(b+c)}{b^3+c^3}+\dfrac{b^3(a+c)}{a^3+c^3}+\dfrac{c^3(b+a)}{b^3+a^3} \geq a+b+c \iff a^9b+a^9c+b^9a+b^9c+c^9a+c^9b \ge a^7b^3+a^7c^3+b^7a^3+b^7c^3+c^7a^3+c^7b^3$
lemma: $a^n+b^n\ge a^{n-1}b+b^{n-1}a \ge a^{n-2}b^2+b^{n-2}a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/434166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Finding the value of $\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$ Is it possible to find the value of
$$\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$$
Does it help if I set it equal to $x$? Or I mean what can I possibly do?
$$x=\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$$
$$x^2=1+2\sq... | In the spirit of Ramanujan, let
$$G(x)=\sqrt{x-1+x\sqrt{(x+n-1)+(x+n)\sqrt{(x+2n-1)+(x+2n)\sqrt{\ldots}}}}$$
and note that by setting $x=2$ and $n=1$ we recover our nested radical. After squaring,
$$G(x)^{2}=(x-1)+xG(x+n)$$
This is a slightly more complicated functional than Ramanujan encountered and I'm not sure how t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
"answer_count": 5,
"answer_id": 1
} |
How to solve these three equations? If α ,β ,γ are three numbers s.t.:
$\ α^ \ $ + $\ β \ $ + $ γ \ $ = −2
$\ α^2 \ $ + $\ β^2 \ $ + $ γ^2 \ $ = 6
$\ α^3 \ $ + $\ β^3 \ $ + $ γ^3 \ $ = −5,
then $\ α^4 \ $ + $\ β^4 \ $ + $ γ^4 \ $ is equal to ??
I tried out substituting the values of each equation to one other .... | Let $A_{n}=a^n+b^n+c^n$. Then we have
$$A_{n+3}=(a+b+c)A_{n+2}-(ab+bc+ac)A_{n+1}+abcA_{n}$$
and
$$2(ab+bc+ac)=(a+b+c)^2-(a^2+b^2+c^2)=4-6=-2$$
$$a^3+b^3+c^3=(a+b+c)^3-3(a+b+c)(ab+bc+ac)+3abc$$
Then we can easily find $abc$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/435982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
show that the function $z = 2x^2 + y^2 +2xy -2x +2y +2$ is greater than $-3$ Show that the function
$$z = 2x^2 + y^2 +2xy -2x +2y +2$$
is greater than $-3$
I tried to factorize but couldn't get more than $(x-1)^2 + (x+y)^2 +(y-1)^2 - (y)^2$.
Is there any another way to factorize or another method??
| Here is a way of doing it that requires no previous knowledge of 3D analytic geometry or calculus:
You are trying to show $z = 2x^2+y^2+2xy-2x+2y+2 \gt -3$, or, (more usefully) that $z+3 = 2x^2+y^2+2xy-2x+2y+5 \gt 0$
$$z+3 = 2x^2+y^2+2xy-2x+2y+5 \\= 2x^2+(2y-2)x+(y^2+2y+5)$$
Now we can use the quadratic formula on it a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/436679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Factor Equation Help me with this,
Question: factor $x^3y-x^3z+y^3z-xy^3+xz^3-yz^3$.
Solution:
$$\begin{eqnarray}&=&x^3y-x^3z+y^3z-xy^3+xz^3-yz^3\\
&=&x\left(z^3-y^3\right)+y\left(x^3-z^3\right)+z\left(y^3-x^3\right)\\
&=&x\left[(z-y)\left(z^2+zy+y^2\right)\right]+y\left[(x-z)\left(x^2+xz+z^2\right)\right]+z\left[(y-x)... | If $E(x,y,z)=x^3y-x^3z+y^3z-xy^3+xz^3-yz^3=x^3(y-z)+y^3(z-x)+z^3(x-y)$
Observe that $E=0$ if $x=y\implies x-y$ divides $f$
Similarly, $y-z,z-x$ divide $f$ $\implies E=(x-y)(y-z)(z-x)F$
As $E$ is Alternating polynomial wrt $x,y,z$
and "the ratio of two alternating polynomials is a symmetric function" (Source : wiki, P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
$a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0 $ Let $a,\,b,\,c,\,d$ be distinct real numbers and $a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0$. Then ... | Let the two polynomials be $$
p(x) = x^2 - 2cx -5d \\
q(x) = x^2 - 2ax - 5b
$$
You also know that $$
p(x) = (x-a)(x-b) = x^2 - (a+b)x + ab \\
q(x) = (x-c)(x-d) = x^2 - (c+d)x + cd
$$
You might try playing around with these two forms. For example, you can take the product of the polynomials and equate the coefficients ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Prove that $n^k < 2^n$ for all large enough $n$ If $k\ge 2$ is an integer,
prove by elementary means
(no calculus or limits)
that
there is a $N(k)$
such that
$n^k < 2^n$
for all integers $n \ge N(k)$.
Give an explicit form for $N(k)$.
| Write the inequality as
$n < 2^{n/k}$.
Suppose this is true for $n$.
I want to find conditions such that
this is also true for $n+1$.
By assumption,
$n+1
= n(1+\frac1{n})
< 2^{n/k}(1+\frac1{n})
$
and this is
less than
$2^{(n+1)/k}$
if
$2^{n/k}(1+\frac1{n})
< 2^{(n+1)/k}$
or
$2^{1/k} > 1+\frac1{n}$.
I showed in another... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/439026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
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.