Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Conjugacy classes in a matrix group Consider the matrix group $PGL_{2}(\mathbb{F}_{q})$ for $q$ odd. Why is it that $\begin{pmatrix} -1 & 0\\ 0 & 1\end{pmatrix}$ has $q(q + 1)/2$ elements in its conjugacy class while $\begin{pmatrix} 2 & 0\\ 0 & 1\end{pmatrix}$ has $q(q + 1)$ elements?
| The centralizer of $\begin{pmatrix} -1 & 0\\ 0 & 1\end{pmatrix}$ in $PGL(2,q)$ is given by all matrices $\begin{pmatrix} a & b\\ c & d\end{pmatrix}$, such that
$$
\begin{pmatrix} -1 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix} a & b\\ c & d\end{pmatrix}\begin{pmatrix} -1 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} ka & k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Simplifying an expression $\frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)}$ if we know $x+y+z=0$ The following expression is given:
$$\frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)}$$
Simplify it, knowing that $x+y+z=0$.
| Use that $z=-(x+y)$, so you have that the numerator turns out to be $$x^7+y^7-(x+y)^7=-7x^6y-21x^5y^2-35x^4y^3-35x^3y^4-21x^2y^5-7xy^6$$Also, the denominator turns out to be $$xy(-x-y)(x^4+y^4+(-x-y))^4=-2x^6y-6x^5y^2-10x^4y^3-10x^3y^4-6x^2t^5-2xy^6$$You can factor a $7$ from the numerator and a $2$ from the denominato... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Find the minimum value.
Find the minimum value of $4^x + 4^{1-x}$ , $x\in\mathbb{R}$.
In this I used the property that $a + \frac{1}{a}\geq 2$.
So I begin with
$$ 4^x + \left(\frac{1}{4}\right)^x + 3\left(\frac{1}{4}\right)^x \geq 2 + 3\left(\frac{1}{4}\right)^x$$
So I think the minimum value be between 2 to 3.
Bu... | Your approach was a good one, except for the lack of "$x$, $-x$" symmetry.
But that is not hard to fix. Since $1/2$ is the midway point between $0$ and $1$, it seems natural to look instead at $4^{x-1/2}+4^{1/2-x}$. This is a close relative of our expression, since
$$4^x+4^{1-x}=4^{1/2}(4^{x-1/2}+4^{1/2-x}).$$
Now it'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Partial Fraction Decomposition of $\frac{x^4+2}{x^5+6x^3}$ I tried answering the following question but I'm getting it wrong for some reason. I would appreciate any help.
$$\frac{x^4+2}{x^5+6x^3}$$
My answer:
$$\frac{A}{x}+\frac{Bx+C}{x^2}+\frac{Dx+E}{x^3}+\frac{Fx+G}{x^2+6}$$
What am I doing wrong?
| The typical way to deal with $(x+b)^n$ in the denominator is to have the terms
$$ \frac{A_1}{x+b} + \frac{A_2}{(x+b)^2} + \dots + \frac{A_n}{(x+b)^n}$$
Note that $A_2$, $A_3$ etc are constant terms.
In your case, try expressing it in the form
$$\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x^2+6}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/116199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Eigenvalues of a matrix Let $A$ be a square matrix of order, say, $4$.
Consider the matrix $$B=\left( \begin{array}{ccc}A &I \\ I & A\end{array} \right)$$ where $I$ is the identity matrix of order $4$.
Let $\lambda$ be an eigenvalue of $A$. Then there exists a nonzero vector $\bf{x}=$ $\left( \begin{array}{c} x_1 &x_2... | Here is a useful observation. For any $k$, let $I_k$ denote the $k \times k$ identity matrix.
Fixing a positive integer $n$, you can check that the $(2n) \times (2n)$ square matrix
$$
C = \frac{1}{\sqrt{2}} \begin{pmatrix} I_n & -I_n \\ I_n & I_n \end{pmatrix}
$$
satisfies $C^T C = CC^T = I_{2n}$ (here $C^T$ denotes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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How many times do these two graphs intersect for values x >0? When the curves
$$y = x^2 + 4x -5$$
and
$$y = \frac{1}{1+x^2} $$
are drawn in the $xy$-plane, how many times do the two graphs intersect for values of $x > 0$ ?
I equate the value of $y$, then the equation comes in the fourth power of $x$ . How I can solve ... | While polynomials of degree 4 can be solved by radicals, that is not needed here.
The first graph, $y=x^2+4x-5 = (x+5)(x-1)$ is positive if $x\lt -5$ or if $x\gt 1$. It is increasing on $x\gt 1$. The graph of $y=\frac{1}{1+x^2}$ is always positive, and is decreasing on $x\gt 0$.
When we look at the portions of the gra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Maclaurin expansion $\log\left( \frac{1+x}{1-x}\right)$, show equality of two sums I am supposed to find the Maclaurin expantion of
$ \log\left( \frac{1+x}{1-x} \right) $
So I noticed the obvious that $\log(1+x) - \log(1-x)$
Then Maclaurin polynomial of $\log (1+x)$ equals $\sum_{n=0}^{\infty} (-1)^{n+1}\frac{x^n}{n... | The expansion for $\log (1+x)$ is not $\sum_{n=0}^{\infty} (-1)^{n+1}\frac{x^n}{n}$, but
$$\begin{equation*}
\log (1+x)=\sum_{n=0}^{\infty }\frac{\left( -1\right) ^{n}}{n+1}%
x^{n+1}=\sum_{n=1}^{\infty }\frac{\left( -1\right) ^{n+1}}{n}x^{n+1}
\end{equation*}.$$
Consequently
$$
\begin{eqnarray*}
\log (1-x) &=&\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $a^6-1$ is divisible by $168$ whenever $(a,42)=1$. I have been running into this type of problem a lot:
Show that $a^6-1$ is divisible by $168$ whenever $(a,42)=1$.
First of all, by Euler's theorem, we have that
$$a^{\phi(42)}\equiv a^{12}\equiv1\pmod{42}.$$
Notice that
$$a^6a^6\equiv1\pmod{42}\text{ and }1... | By Fermat's Theorem, $a^6\equiv 1 \pmod 7$. Also by Fermat's Theorem, or otherwise, $a^2\equiv 1 \pmod 3$. Thus $a^6\equiv 1 \pmod 3$. So far, we have that
$$a^6\equiv 1 \pmod {21}.$$
But $a$ is odd, so $a^2\equiv 1 \pmod 8$. It follows that
$$a^6 \equiv 1 \pmod {8}.$$
Now it's over.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/117355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Number of integer solutions to $3i^2 + 2j^2 = 77 \cdot 6^{2012}$ here is another problem I did not manage to solve in the contest I mentioned in my previous question.
Determine the number of integer solutions $(i, j)$ of the equation:
$3i^2 + 2j^2 = 77 \cdot 6^{2012}$.
Applying logarithms is not useful, since on th... | $A)~ i^2=25x^2 ~\text {and}~ j^2=x^2$
$3 \cdot 25x^2+2x^2=77 \cdot 6^{2012} \Rightarrow x^2=6^{2012}$ , hence :
$i= \pm 5x \Rightarrow i = \pm 5 \cdot 6^{1006}$
$j= \pm x \Rightarrow j = \pm 6^{1006}$
$R_A :$
$(i,j) \in \{(-5 \cdot 6^{1006},-6^{1006}),(-5 \cdot 6^{1006},6^{1006}),(5 \cdot 6^{1006},-6^{1006}),(5 \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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Show that $\displaystyle{\frac{1}{9}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$ Show that $\displaystyle{\frac{1}{9}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$
Use proof by induction. I tried for $n=1$ and got $\frac{27}{9}=3$, but if I assume for $n$ and show it for $n+1$, I don't know what... | This is the same as proving $(10^n +3⋅4^n +5)$ is divisible by 9. The rule for divisibility by $9$ is if it's digital sum is $9$, then it's divisible by $9$
The sum of the three parts should give a digital sum of $9$.
$10^n$ digital sum $=1$
$4^n$ digital sum repeats in the cycle $(4,7,1)$
multiplying these by 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 6
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Minimum of integral Let $S$ be the set of all integrable on $[0,1]$ such that $$\int\limits_0^1f(x)dx=\int\limits_0^1xf(x)dx+1=3.$$
Prove that $S$ is infinite and evaluate $$\min\limits_{f\in S}\int\limits_0^1f^2(x)dx.$$
| We have $3 = \int_0^1 f(x) \cdot \left(x +\frac{1}{3} \right) \, dx$. Let's use Cauchy–Schwarz inequality:
$$3 = \int_0^1 f(x) \cdot \left(x +\frac{1}{3} \right) \, dx \le \sqrt{ \int_0^1 f^2(x) \, dx} \sqrt{\int_0^1 \left(x +\frac{1}{3} \right)^2 dx} = \sqrt{\frac{7}{9}} \sqrt{ \int_0^1 f^2(x) \, dx}$$
Hence:
$$\int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Gaussian proof for the sum of squares? There is a famous proof of the Sum of integers, supposedly put forward by Gauss.
$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$
$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$
$$S=\frac{n(1+n)}{2}$$
I was looking for a similar proof for when $S=\sum\limits_{i=1}^{n}i^2$
I've tried th... | Since I think the solution Tyler proposes is very useful and accesible, I'll spell it out for you:
We know that
$$(k+1)^3-k^3=3k^2+3k+1$$
If we give the equation values from $1$ to $n$ we get the following:
$$(\color{red}{1}+1)^3-\color{red}{1}^3=3\cdot \color{red}{1}^2+3\cdot \color{red}{1}+1$$
$$(\color{red}{2}+1)^3-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/122546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 7,
"answer_id": 0
} |
Prove for any positive real numbers $a,b,c$ $\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2} \geq \frac{a+b+c}{3}$ Since the problem sheets says I should use Cauchy-Schwarz inequality, I used
$\frac{{a_1}^2}{x_1}+\frac{{a_2}^2}{x_2}+\frac{{a_3}^2}{x_3}$
$\geq \frac{(a_1+a_2+a_3)^2}{x_1+x_2+x_3}$
I... | We need to prove that
$$\sum_{cyc}\left(\frac{a^3}{a^2+ab+b^2}-\frac{a}{3}\right)\geq0$$ or
$$\sum_{cyc}\frac{2a^3-a^2b-ab^2}{a^2+ab+b^2}\geq0$$ or
$$\sum_{cyc}\left(\frac{a(a-b)(2a+b)}{a^2+ab+b^2}-(a-b)\right)\geq0$$ or
$$\sum_{cyc}\frac{(a-b)^2(a+b)}{a^2+ab+b^2}\geq0.$$
Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/122741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Evaluate and prove by induction: $\sum k{n\choose k},\sum \frac{1}{k(k+1)}$
*
*$\displaystyle
0\cdot \binom{n}{0} + 1\cdot \binom{n}{1} + 2\binom{n}{2}+\cdots+(n-1)\cdot \binom{n}{n-1}+n\cdot \binom{n}{n}$
*$\displaystyle\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3}+\frac{1}{3\cdot 4} +\cdots+\frac{1}{(n-1)\cdot n}$
Ho... | If you don't have to use induction, you can do this:
1)
$$
\sum_k k\binom{n}{k} = \sum_k k \frac{n(n-1)\cdots(n-k+1)}{k!} = n\sum_k \binom{n-1}{k-1} = n \sum_k \binom{n-1}{k} = n2^{n-1}
$$
Here $k$ runs over all integers and by convention $\binom{n}{k}$ is defined as zero if $k<0$ or $k > n$. The last equality follow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/123655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 2
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can't figure out what I been doing wrong on simple integration question $$\int_0^1 \sqrt{(\sqrt{5})^2+(2t)^2}\;dt$$
Based on the formula
$\int \sqrt{a^2+x^2}\;dx=\frac{1}{2}[x\sqrt{a^2+x^2}+a^2\log(x+\sqrt{a^2+x^2})]$
I just plug in above input into the formula above
However I can only find $3+\frac{5}{2}\log(5)$
but ... | we have
\begin{align*}
\int_0^1 \sqrt{5 + (2t)^2}\, dt &= \frac 12 \int_0^2 \sqrt{5 + x^2}\, dx\\
&= \frac 14\left[x\sqrt{5 + x^2} + 5\log\bigl(x + \sqrt{5 + x^2}\bigr)\right]_0^2\\
&= \frac 14\left(6 + 5\log 5 - 0 - 5 \log \sqrt 5\right)\\
&= \frac 32 + \frac 58 \, \log 5.
\end{align*}
I don'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to Derive a Double Angle Identity. How does one derive the following two identities:
$$\begin{align*}
\cos 2\theta &= 1-2\sin^2\theta\\
\sin 2\theta &= 2\sin\theta\cos\theta
\end{align*}$$
| Hints:
For the $\cos 2\theta$ formula, use the sum identity (with $x=y=\theta$)
$$
\cos(x+y)=\cos x\cos y - \sin x\sin y,
$$
followed by the Pythagorean identity $\cos^2 x=1-\sin^2 x $.
For the $\sin 2\theta$ formula , use the sum identity (with $x=y=\theta$)
$$
\sin(x+y)=\sin x\cos y +\sin y\cos x.
$$
Or, for the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/126894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Real VS Complex for integrals: $\int_0^\infty \frac{dx}{1 + x^3}$ The integral
$$\int_0^\infty \frac{dx}{1 + x^4} = \frac{\pi}{2\sqrt2}$$
can be evaluated both by a complex method (residues) and by a
real method (partial fraction decomposition).
The complex method works also for the integral
$$\int_0^\infty \frac{dx}{... | Note that for $a > 0$, $$\int_0^N \frac{1}{x+a}\ dx = \ln(N+a) - \ln(a) = \ln(N) - \ln(a) + o(1)\ \text{as} \ N \to \infty$$ while
$$\eqalign{\int_0^N \frac{x+a}{(x+a)^2 + b^2}\ dx &= \frac{1}{2} \left(\ln((N+a)^2+b^2) - \ln(a^2+b^2)\right)\cr &= \ln(N) - \ln(a^2+b^2) + o(1) \ \text{as} \ N \to \infty\cr}$$
and (if $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
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Graphing simple fractional function I am trying to graph
$$f(x) = \frac{x^2-4}{x^2+4}$$
It seems pretty simple to me but I can't finish it correctly.
I know that there is a horizontal asymptote at $1$ beacuse the degrees on the variable at the same and $\frac{x}{x}$ is 1.
I know that it is a negative function until ze... | Correct second derivative:
$$\begin{align*}
f(x) &= \frac{x^2-4}{x^2+4}\\
f'(x) &= \frac{(x^2+4)(x^2-4)' - (x^2-4)(x^2+4)'}{(x^2+4)^2}\\
&= \frac{(x^2+4)(2x) - (x^2-4)(2x)}{(x^2+4)^2}\\
&= \frac{2x(x^2+4-x^2+4)}{(x^2+4)^2}\\
&= \frac{16x}{(x^2+4)^2}.\\
f''(x) &= \frac{(x^2+4)^2(16x)' - 16x\Bigl( (x^2+4)^2\Bigr)'}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/128827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Maximize volume of box in ellipsoid I need to find the dimensions of the box with maximum volume (with faces parallel to the coordinate planes) that can be inscribed in ellipsoid
$$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$$
A hint given was:
If vertex of box in first octants is (x,y,z) then volume is 8xyz.... | You can cheat here, because the property of being a maximum-volume axis-parallel box is preserved by "stretching" transformations of the form $(x,y,z) \mapsto (ax,by,cz)$ (because such a transformation preserves the property of being axis-parallel, and multiplies all volumes by the constant $abc$).
So start with a sphe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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How to compute $\int{\frac{5x^3+8x^2+x+2}{x^2(2x^2+1)}} dx$? $$\int{\frac{5x^3+8x^2+x+2}{x^2(2x^2+1)}} dx$$
So ... how do I start? Numerator cant be factorized it seems, and this looks like a complicated expression ...
I tried expanding the denominator to see if integration by substitution will work, but it didn't giv... | If you wish to avoid partial fractions, it is possible to clean up that denominator with a simple substitution. To make that $2x^2+1$ factor more manageable, make the substitution
$x=\frac{\sqrt2}2\tan\theta,dx=\frac{\sqrt2}2\sec^2\theta d\theta$
$\int\dfrac{5x^3+8x^2+x+2}{x^2(2x^2+1)}dx=\int\dfrac{\frac{5\sqrt2}4\tan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
$3^k$ not congruent to $-1 \pmod {2^e}, e > 2$. $3^k \not\equiv -1 \pmod {2^e}$ for $e > 2, k > 0$. Is this true? I have tried to prove it by expanding $(1 + 2)^k$. [Notation: $(n; m) := n! / (m! (n - m)!)$] E.g., for $e = 3$ I get: $(1+2)^k + 1 = 2 + (k; 1) 2 + (k; 2) 2^2 + (k; e) 2^e + ...$ So, here it's enough to pr... | Based on pedja,we only prove that $8 \nmid 3^k+1$,
1.when $k=2m+1$,$3^k+1=3^{2m+1}+1=9^m \times 3+1 \equiv 4 \pmod{8} $
2.when $k=2m$,$3^k+1=3^{2m}+1=9^m+1 \equiv 2 \pmod{8} $
so we have $8 \nmid 3^k+1$,this is
$3^k \not\equiv -1 \pmod {2^e}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/130814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Differentiation of $y = \tan^{-1} \Bigl\{ \frac{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+ \sqrt{1-x^{2}}}\Bigr\}$ How do i differentiate the following: $$y = \tan^{-1} \biggl\{ \frac{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+ \sqrt{1-x^{2}}}\biggr\}$$
I know that $\text{derivative}$ of $\tan^{-1}{x}$ is $\... | Putting $x^2=\cos2z,1+x^2=1+\cos2z=2\cos^2z,1-x^2=1-\cos2z=2\sin^2z$
$$\frac{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+ \sqrt{1-x^{2}}}=\frac{\cos z-\sin z}{\cos z+\sin z}=\frac{1-\tan z}{1+\tan z}=\tan\left(\frac\pi4-z\right)$$
$$y = \tan^{-1} \biggl\{ \frac{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+ \sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I solve $(x-1)(x-2)(x-3)(x-4)=3$ How to solve $$(x-1) \cdot (x-2) \cdot (x-3) \cdot (x-4) = 3$$
Any hints?
| Hint $\ $ The LHS is a difference of squares $\rm\:y^2\!-\!1,\:$ hence so too is $\rm\:(y^2\!-\!1)-3\: =\: y^2\!-\!2^2,\:$ viz.
$\rm\qquad\ \:\! (x\!-\!1)(x\!-\!4) (x\!-\!2)(x\!-\!3)\ =\ (x^2\!-\!5x+4)(x^2\!-\!5x+6)\ =\ (x^2\!-\!5x+5)^2 \!-\! 1^2 $
$\rm\ \ \Rightarrow\ (x\!-\!1)(x\!-\!4) (x\!-\!2)(x\!-\!3)\!-\!3\ =\ (x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/132572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Another way to go about proving Binet's Formula As I showed in another question of mine, it is easy to prove that
$$\tag{1}\phi^{n+1} =F_{n+1} \phi+F_{n }$$
given $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}\text{ ; }n\geq2$.
Now, extending $(1)$ to negative and zero indices naturally yields:
$$\eqalign{
& {F_{ 0}} = 0 ... | The set of sequences $x_n$ that have the property that $x_{n+2} = x_{n+1} + x_n,$ with, say, real number values, makes a vector space over the reals of dimension 2. Taking the two roots of $\lambda^2 = \lambda + 1,$ the larger being
$$ \phi = \frac{1 + \sqrt 5}{2} $$ and the smaller being $-1 / \phi,$ any such sequenc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Complex Series question Would you please give hint for these two(not homework)~~ $$\Pi_{n=2}^{\infty}(1-\frac{1}{n^2})=\frac{1}{2}$$ and $$\Pi_{1}^{\infty}(1+\frac{}{})e^{\frac{-z}{n}} \text{ converges absolutely and uniformly on every compact set }$$
| For $(1)$, note that
$$1-\frac{1}{n^2}=\frac{n^2-1}{n^2}=\frac{n-1}{n}\frac{n+1}{n}.$$
Now write down the first few terms of the product, in factored form. (So each term of the original product is represented as the product of two terms.) We get
$$\frac{1}{2}\frac{3}{2}\frac{2}{3}\frac{4}{3}\frac{3}{4}\frac{5}{4}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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More highschool math $\frac{3^n(n^3+3n^2+3n+1)}{3^{n+1}\cdot n^3} = \frac{n^3+3n^2+3n+1}{3n^3} \to \frac{1}{3}$ So the question I am trying to work through is:
Test the series
$$\frac{1}{3}+\frac{2^3}{3^2}+\frac{3^3}{3^3}+\frac{4^3}{3^4}+\frac{5^3}{3^5}+\cdot\cdot\cdot$$
for convergence.
The solution (using D'Alember... |
What happens with the $3^n$ in the numerator and the $3^{n+1}$ in the denominator?
Recall the following laws of exponents:
$$a^{b}a^c = a^{b+c}, \quad \text{ and } \quad \frac{a^b}{a^c} = a^{b-c} = \frac{1}{a^{c-b}},$$
for any $a>0$, and any real numbers $b,c$. In particular, if $a=3$, $b=n$, and $c=n+1$, then:
$$\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Determine convergence of $\sum_{n=1}^{\infty} (\cos{\frac{2}{n}}-\cos{\frac{4}{n}})$ Determine convergence of $$\sum_{n=1}^{\infty} \left(\cos{\frac{2}{n}}-\cos{\frac{4}{n}}\right)$$
In the answer, it says
$$\cos{\frac{2}{n}}-\cos{\frac{4}{n}} = 2\sin{\frac{3}{n}}\sin{\frac{1}{n}} \le 2\cdot \frac{3}{n} \cdot \frac{1}... | I suggest another way:
$\cos\left(\frac{2}{n}\right)-\cos\left(\frac{4}{n}\right)=-\left(1-\cos\left(\frac{2}{n}\right)\right)+1-\cos\left(\frac{4}{n}\right)\sim -\frac{4}{2n^2}+\frac{16}{2n^2}=\frac{6}{n^2}$ and $\sum \frac{6}{n^2}$ converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/139272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Polynomial regression interpolation?
Possible Duplicate:
Writing a function $f$ when $x$ and $f(x)$ are known
I'm not versed in mathematics, so you'll have to speak slowly...
If I want to fit a curve to the points,
X Y
1 0.5
2 5.0
3 0.5
4 2.5
5 5.0
6 0.5
Where would I begin? For my purposes, this needs to be... | Here is a step by step way of finding a polynomial $f(x)$ such that for $1 \leq i \leq 6$, $f(i)$ has the values you stated. We will "build up" $f(x)$ slowly, calling the intermediate stages $f^{(1)}(x), f^{(2)}(x), \ldots $ and so on.
*
*Set $y = f^{(1)}(x) = 0.5$ where the right side doesn't really depend on $x$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/145584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to find limit of function: $\lim\limits_{n \to \infty} \frac{\sqrt{n}}{2} \arccos(\frac{n-2}{22+n})$ How would I find this limit?
$\lim_{n \to \infty} \frac{\sqrt{n}}{2} \bigl(\arccos(\frac{n-2}{22+n}))$
| As the argument of $\arccos$ approaches 1 for large $n$, the inverse cosine function approaches zero. Rewriting the expression as quotient of two expression approaching zero for
large $n$
$$
\frac{ \arccos\left( \frac{1-2/n}{1+22/n} \right)}{2 n^{-1/2}}
$$
we can apply the L'Hospital's rule:
$$ \begin{eqnarray}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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} |
Minimal polynomial of the root of algebraic number I have obtained the minimal polynomial of $9-4\sqrt{2}$ over $\mathbb{Q}$ by algebraic operations:
$$ (x-9)^2-32 = x^2-18x+49.$$
I wonder how to calculate the minimal polynomial of $\sqrt{9-4\sqrt{2}}$ with the help of this sub-result? Or is there a smarter way to do t... | $\rm x^4-18\,x^2+49\:$ will be the minimal polynomial, unless $\rm\:\sqrt{9-4\sqrt{2}}\:$ denests to $\rm\:a + b\sqrt{2}.\:$
This can be tested by a radical denesting formula that I discovered as a teenager.
Simple Denesting Rule $\rm\ \ \ \color{blue}{subtract\ out}\ \sqrt{norm}\:,\ \ then\ \ \color{brown}{divide\ o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
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Calculating the shortest possible distance between points Question:
Given the points $A(3,3)$, $B(0,1)$ and $C(x,0)$ where $0 < x < 3$, $AC$ is the distance between $A$ and $C$ and $BC$ is the distance between $B$ and $C$. What is x for the distance $AC + BC$ to be minimal?
What have I done?
I defined the function $AC ... | The following little trick (method?) is useful in many places. Let $y=3-x$. We want to minimize $\sqrt{1+x^2}+\sqrt{3^2+y^2}$, where $x+y=3$.
Differentiate with respect to $x$. Using the fact that $\frac{dy}{dx}=-1$, we arrive at the equation
$$\frac{x}{\sqrt{1+x^2}}=\frac{y}{\sqrt{9+y^2}}.$$
Note that this is (apart... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
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Integral of $\int \frac{5x+1}{(2x+1)(x-1)}$ I am suppose to use partial fractions
$$\int \frac{5x+1}{(2x+1)(x-1)}$$
So I think I am suppose to split the top and the bottom. (x-1)
$$\int \frac{A}{(2x+1)}+ \frac{B}{x-1}$$
Now I am not sure what to do.
| But $$\frac{5x+1}{(2x+1)(x-1)}\ne\frac{5x+1}{2x+1}+\frac{5x+1}{x-1}\;,$$ as you’ll see if you combine the fractions on the righthand side over a common denominator: you get
$$\begin{align*}
\frac{5x+1}{2x+1}+\frac{5x+1}{x-1}&=\frac{5x+1}{2x+1}\cdot\frac{x-1}{x-1}+\frac{5x+1}{x-1}\cdot\frac{2x+1}{2x+1}\\\\
&=\frac{(5x+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Arc length of $y = \frac{x^3}{3} + \frac{1}{4x}$ Arc length of $y = \frac{x^3}{3} + \frac{1}{4x}$ over $1 \leq x \leq 2$
I know that the first thing I need to do is take the derivative.
$$y' = x^2 - 4x^{-2}$$
Then I take the integral on that range using the arc length formula.
$$\int_1^2 \sqrt{1 + (x^2-4x^{-2})^2}$$
$$... | Your derivative is wrong, and your squaring is wrong.
*
*Your derivative is wrong:
$$\left(\frac{x^3}{3} +\frac{1}{4x}\right)' = \left(\frac{1}{3}x^3 + \frac{1}{4}x^{-1}\right)' = x^2 - \frac{1}{4}x^{-2} = x^2 - \frac{x^{-2}}{4}.$$
*You squared incorrectly: if you square $x^2-4x^{-2}$, you get:
$$(x^2-4x^{-2})^2 = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is wrong with my solution? $\int \cos^2 x \tan^3x dx$ I am trying to do this problem completely on my own but I can not get a proper answer for some reason
$$\begin{align}
\int \cos^2 x \tan^3x dx
&=\int \cos^2 x \frac{ \sin^3 x}{ \cos^3 x}dx\\
&=\int \frac{ \cos^2 x\sin^3 x}{ \cos^3 x}dx\\
&=\int \frac{ \sin^3 ... | The calculation is correct. There are many alternate forms of the integral, because of the endlessly many trigonometric identities.
If you differentiate the expression you got and simplify, you will see that you are right.
The answer you saw is likely also right. What was it?
Added: Since $\sec x=\frac{1}{\cos x}$, w... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
"answer_id": 2
} |
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$ Compute
$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
| Consider:
$$I(a) = \int_0^1 \frac{\ln (1+ax)}{1+x^2} \, dx$$
than, the derivative $I'$ is equal:
$$I'(a) = \int_0^1 \frac{x}{(1+ax)(1+x^2)} \, dx = \frac{2 a \arctan x - 2\ln (1+a x) + \ln (1+x^2)}{2(1+a^2)} \Big|_0^1\\
= \frac{\pi a + 2 \ln 2}{4(1+a^2)} - \frac{\ln (1+a)}{1+a^2}$$
Hence:
$$I(1) = \int_0^1 \left( \fra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "102",
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"answer_id": 0
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When does $\sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n$ converge? For what values of $z \in \mathbb{C}$ does the following series converge:
$$\displaystyle \sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n\quad ?$$
| You're given
$$f(z)=\displaystyle \sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n$$
A sensible solution would be using Cauchy's Root test. We want to find
$$\lim\limits_{n\to\infty}\left(\frac{2^n+n^2}{3^n+n^3}\right)^{1/n} =$$
$$=\lim\limits_{n\to\infty}\frac 2 3\left(\frac{1+n^2/2^n}{1+n^3/3^n}\right)^{1/n} =$$
$$=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Equivalent of $ I_{n}=\int_0^1 \frac{x^n \ln x}{x-1}\mathrm dx, n\rightarrow \infty$ I would like to show that
$$ I_{n}=\int_0^1 \frac{x^n \ln x}{x-1}\mathrm dx \sim_{n\rightarrow \infty} \frac{1}{n}$$
Using the change of variable $u=x^n$:
$$ I_{n}=\frac{1}{n^2} \int_0^1 \frac{u^{1/n} \ln u}{u^{1/n}-1} \mathrm du=\frac... | $$\frac{x^n}{x-1} = \frac{x^n-1}{x-1} + \frac{1}{x-1}= \left(1+ x+ \cdots + x^{n-1}\right) +\frac{1}{x-1}.$$
So $$I_n = \int^1_0 (1+x+\cdots + x^{n-1}) \log x + \int^1_0 \frac{\log x}{x-1} dx.$$
Integration by parts shows $ \int^1_0 x^k \log x = -1/(k+1)^2$ and expanding $\log$ by Taylor series will show $\displaystyle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Another Congruence Proof I've been asked to attempt a proof of the following congruence. It is found in a section of my textbook with Wilson's theorem and Fermat's Little theorem. I've pondered the problem for a while and nothing interesting has occurred to me.
$1^23^2\cdot\cdot\cdot(p-4)^2(p-2)^2\equiv (-1)^{(p+1)/2... | An idea that, perhaps, will appeal to you besides the ones already given above:$$1^23^2\cdot\ldots\cdot (p-1)^2=\frac{\left(1\cdot 2\cdot\ldots\cdot (p-1)\right)^2}{\left(2\cdot 4\cdot\ldots\cdot (p-1)\right)^2}\,\,(**)$$Now we use Wilson's theorem, Fermat's Little Theorem and arithmetic modulo $p$: $$(**)\,\,=\frac{(-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Limit involving $(\sin x) /x -\cos x $ and $(e^{2x}-1)/(2x)$, without l'Hôpital Find:
$$\lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$$
I have factorized it in this manner in an attempt to use the formulae.
I have tried to use that for $x$ tending to $0$, $\dfrac{\sin x}{... | You are given
$$\lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$$
I guess you know
$$\lim_{x\to 0}\dfrac{\sin x}{x}=1$$
$$\lim_{x\to 0} \dfrac{e^{2x} - 1}{2x}=1$$
The most healthy way of solving this is using
$$\frac{\sin x}{x} = 1-\frac {x^2}{6}+o(x^2)$$
$$\frac{e^x-1}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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Solving a differential equation I am trying to find the solution of the equation
t $y''-(\cos x) y'+(\sin x )y = 0$.
I need help urgently.Thanks
| This is a linear ODE of trigonometric function coefficients. The current approach of solving it is to transform it to a linear ODE of polynomial function coefficients first.
Let $u=\sin x$ ,
Then $\dfrac{dy}{dx}=\dfrac{dy}{du}\dfrac{du}{dx}=(\cos x)\dfrac{dy}{du}$
$\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\left((\cos x)\dfrac{d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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The rank of a multiplication map $L\colon M_{2 \times 3} \to M_{3 \times 3}$
Let $L\colon M_{2 \times 3} \to M_{3 \times 3}$ be the linear transformation defined by $L(A) = \left[\begin{array}{rr}2 & -1 \\ 1 & 2 \\ 3 & 1\end{array}\right]\,A$. Find the dimension of the range of $L$.
Answer: $6$
How is the answer $6$?... | If we identify $M_{2 \times 3}$ with $\Bbb{R}^6$ and similarly with the other vector space, we see that $L$ is a linear transformation from $\Bbb{R}^6$ to $\Bbb{R}^9$. Now I claim that the kernel of $L$ is trivial. Indeed, suppose there is a matrix
$$A = \left[\begin{array}{ccc} a & b & c & \\ d & e & f \end{array}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Doubt on displacement of a parabola(Again) In another exercise is given:
Find the parabola which is a displacement of $y = 2x^2 - 3x + 4$ which passes though the point $(2, -1)$ and has $x = 1$ as its symmetry axis.
I've reduced the based equation to the form: $y = 2(x - \frac{3}{4})^2 + \frac{23}{8}$, so the vertex is... | Your equation for the original parabola should read $y = 2\left(x-\dfrac{3}{4}\right)^2+\dfrac{23}{8}$. The displaced parabola will have the equation $y = 2(x-x_v)^2 + y_v$, where $(x_v, y_v)$ is the vertex. Since the axis is $x=1$, $x_v$ is $1$. The point $(2,-1)$ is on the parabola, so we have $-1 = 2(2-1)^2 + y_v \R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Simplest method to find $5^{20}$ modulo $61$ What is the simplest method to go about finding the remainder of $5^{20}$ divided by $61$?
| Note that $5^3 = 125 = 2(61)+3$. So $5^{20} = (5^3)^6\times 5^2\equiv 3^6\times 5^2 \pmod{61}$.
Now, $3^3\equiv 27$, $3^3\times 5 = 135 \equiv 13\pmod{61}$. So
$$5^{20}\equiv (5^3)^6\times 5^2\equiv 3^6\times 5^2 =(3^3\times 5)^2 \equiv 13^2 \equiv 169\pmod{61}$$
and since $169 = 2(61) + 47$, we finally have that $5^{2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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$|2^x-3^y|=1$ has only three natural pairs as solutions Consider the equation $$|2^x-3^y|=1$$ in the unknowns $x \in \mathbb{N}$ and $y \in \mathbb{N}$. Is it possible to prove that the only solutions are $(1,1)$, $(2,1)$ and $(3,2)$?
| Case 1: If $2^x=3^y+1$ then $x \ge 1$. If $y=0$ then $x=1$.
If $y \ge 1$ then $3^y+1 \equiv 1 \pmod{3}$. Therefore $2^x \equiv 1 \pmod{3}$. Hence $x=2x_1$ with $x_1 \in \mathbb{N}^*$. The equation is equivalent to $$3^y= (2^{x_1}-1)(2^{x_1}+1)$$
Since $2^{x_1}+1>2^{x_1}-1$ and $\gcd (2^{x_1}-1,2^{x_1}+1) \ne 3$ then $2... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Show $\lim_{N\to \infty}\sum_{k=1}^{N}\frac{1}{k+N}=\ln(2)$ I have some difficulty to prove the following limit:
$$\lim_{N\to \infty}\sum_{k=1}^{N}\frac{1}{k+N}=\ln(2)$$
Can someone help me? Thanks.
| $$f(x)=\lim_{n\to\infty} \sum \limits_{k=1}^n \frac{1}{k+\frac{n}{x}}$$
You are looking for $f(1)$
$$f(x)=\lim_{n\to\infty}\frac{x}{n} \sum \limits_{k=1}^n \frac{1}{1+\frac{kx}{n}}=\lim_{n\to\infty} \frac{x}{n} \sum \limits_{k=1}^n (1-\frac{kx}{n}+\frac{k^2x^2}{n^2}-\frac{k^3x^3}{n^3}+....)=\lim_{n\to\infty} \frac{x}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Some method to solve $\int \frac{1}{\left(1+x^2\right)^{2}} dx$ and some doubts. First approach.
$\int \frac{1}{1+x^2} dx=\frac{x}{1+x^2}+2\int \frac{x^2}{\left(1+x^2\right)^2} dx=\frac{x}{1+x^2}+2\int \frac{1}{1+x^2}dx-2\int \frac{1}{\left(1+x^2\right)^2}dx$
From this relationship, I get:
$2\int \frac{1}{\left(1+x^2\... | This integral can be evaluated in that way too as shown below :
$$\int \frac{1}{a^2+x^2} dx=\frac{1}{a} \arctan \frac{x}{a} +c(a) $$
$$\frac {d}{da} (\int \frac{1}{a^2+x^2} dx)= \frac {d}{da}(\frac{1}{a} \arctan \frac{x}{a} +c(a) ) $$
$$\int \frac{-2a}{(a^2+x^2)^2} dx= \frac{-1}{a^2} \arctan \frac{x}{a} + \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170207",
"timestamp": "2023-03-29T00:00:00",
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Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers I have tried the arithmetic-geometric inequality on $(-a+b+c)(a-b+c)(a+b-c)$ which gives
$$(-a+b+c)(a-b+c)(a+b-c) \leq \left(\frac{a+b+c}{3}\right)^3$$
and on $abc$ which gives
$$abc \leq \left(\frac{a+b+c}{3}\right)^3.$$
Since ... | Case 1. If $a,b,c$ are lengths of triangle.
Since
$$
2\sqrt{xy}\leq x+y\qquad
2\sqrt{yz}\leq y+z\qquad
2\sqrt{zx}\leq z+x
$$
for $x,y,z\geq 0$, then multiplying this inequalities we get
$$
8xyz\leq(x+y)(y+z)(z+x)
$$
Now substitute
$$
x=\frac{a+b-c}{2}\qquad
y=\frac{a-b+c}{2}\qquad
z=\frac{-a+b+c}{2}\qquad
$$
Since $a,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 7,
"answer_id": 4
} |
Trigonometric Identities To Prove
*
*$\tan\theta+\cot\theta=\dfrac{2}{\sin2\theta}$
Left Side:
$$\begin{align*}
\tan\theta+\cot\theta={\sin\theta\over\cos\theta}+{\cos\theta\over\sin\theta}={\sin^2\theta+\cos^2\theta\over\cos\theta\sin\theta}
= \dfrac{1}{1\sin\theta\cos\theta}
\end{align*}$$
Right Side:
$$\begin{a... | $$\tan(\theta) + \cot(\theta) = {\sin(\theta)\over \cos(\theta)} + {\cos(\theta)\over \sin(\theta)} = {\sin^2(\theta) + \cos^2(\theta) \over\cos(\theta)\sin(\theta)}
= {1\over\sin(\theta)\cos(\theta)}.$$
Now avail yourself of the fact that $$\sin(2\theta) = 2\cos(
\theta)\sin(\theta).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/170951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to evaluate $\int_{-\infty}^\infty \frac{\cos x}{\cosh x}\,\mathrm dx$ by hand How can I evaluate$$\int_{-\infty}^\infty \frac{\cos x}{\cosh x}\,\mathrm dx\text{ and }\int_0^\infty\frac{\sin x}{e^x-1}\,\mathrm dx.$$
Thanks in advance.
| Let us start from the Weierstrass product for the sine function:
$$ \sin(x) = x \prod_{n\geq 1}\left(1-\frac{x^2}{\pi^2 n^2}\right)\tag{1} $$
This identity over the real line can be proved by exploiting Chebyshev polynomials, but it holds over $\mathbb{C}$ and it is equivalent to
$$ \sinh(x) = x \prod_{n\geq 1}\left(1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 2
} |
Value of $P(12)+P(-8)$ if $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$, $P(1)=10$, $P(2)=20$, $P(3)=30$
What will be the value of $P(12)+P(-8)$ if $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$
provided that $P(1)=10$, $P(2)=20$, $P(3)=30$?
I put these values and got three simultaneous equations in $a, b, c, d$. What is the smarter way to app... | I'm not sure this is the smartest way, but here's one way.
Define $Q(x) = P(x) - x^4$ which satisfies the conditions $$Q(1) = 9, Q(2) = 4, Q(3) = -51.$$
So $a,b,c,d$ satisfy the matrix identity
$$\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \end{pmatrix} \begin{pmatrix} d \\ c \\ b \\ a \end{pmatrix... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
} |
Solving $E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$ $$E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$$
I got no idea how to find the solution to this. Can someone put me on the right track?
Thank you very much.
| $$\text{Let us check the value of }\frac a{\sin\theta}+\frac b{\cos\theta}$$
$$\frac a{\sin\theta}+\frac b{\cos\theta}=\frac{a\cos\theta+b\sin\theta}{\cos\theta\sin\theta}$$
Putting $a=r\sin\alpha,b=r\cos\alpha$ where $r>0$
Squaring & adding we get $r^2=a^2+b^2\implies r=+\sqrt{a^2+b^2}$
$$\implies \frac a{\sin\theta}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Laplace transform of a product of Modified Bessel Functions Working with a scalar field in 2 dimensions I've come to the following integral, from which I can extract the proper ultraviolet behavior ($a \ll 1$) of the theory:
$\int_0^\infty e^{-(4+a^2)x}\left[I_0(2x)\right]^2 ds$.
It is obvious to me that this is the La... | Given that $I_0(2 x)^2 = 1 + 2 x^2 + \frac{3}{2} x^4 + \frac{5}{9}x^6 + \mathcal{o}(x^6)$ we see that it is a hypergeometric function:
$$
I_0(2x)^2 = {}_1F_2\left(\frac{1}{2}; 1,1; 4 x^2\right) = \sum_{n=0}^\infty \frac{\left(\frac{1}{2}\right)_n}{(1)_n (1)_n} \frac{(4 x^2)^n}{n!} = \sum_{n=0}^\infty \left(\frac{x^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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For which angles we know the $\sin$ value algebraically (exact)? For example:
*
*$\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$
*$\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$
*$\sin(30^\circ) = \frac{1}{2}$
*$\sin(45^\circ) = \frac{1}{\sqrt{2}}$
*$\sin(67 \frac{1}{2}^\circ) = \sqrt{ \frac{\sqrt... | Algebraically exact value of Sine for all integer angles is possible. Please visit https://archive.org/details/ExactTrigonometryTableForAllAnglesFinal for the list of exact values for Sine of integer angles in degrees.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/176889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 4
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How do I proceed with these quadratic equations? The question is
$$ax^2 + bx + c=0 $$ and $$cx^2+bx+a=0$$ have a common root, if $b≠ a+c$, then what is $$a^3+b^3+c^3$$
| If a=c, the equations become identical, we can hardly determine any relationship among a,b,c(=a).
If a≠c,
if y is a root of $ax^2+bx+c=0$, then observe that $\frac{1}{y}$ is a root of $cx^2+bx+a=0$.
For the common root, $y=\frac{1}{y}=>y=±1$, but y=-1 makes a-b+c=0 ⇔ b=a+c which is not acceptable according to the give... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Proving $\sin A + \sin B + \sin C = 4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$
Possible Duplicate:
Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
Prove trigonometry identity?
If $A$, $B$, and $C$ are to be taken as the angles of a triangle, then I beg ... | $$
\begin{align}
\sin(A)+\sin(B)+\sin(C)
&=\sin(A)+\sin(B)+\sin(\pi-A-B)\\[9pt]
&=\color{#C00000}{\sin(A)+\sin(B)}+\color{#00A000}{\sin(A+B)}\\[6pt]
&=\color{#C00000}{2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)}+\color{#00A000}{2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A+B}{2}\right)}\\
&=2\sin\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
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Using a circular contour integral I was having some problems preparing for an exam, and a friend of mine told me about this site :)
I have to prove this:
$$
\int_0^{2\pi} \frac{d\theta}{a + \cos\theta} = \frac{2\pi}{\sqrt{a^2 - 1}}
$$
Using
$$
z = e^{i\theta}\\
a>1
$$
and integrating over the unit circle $|z| = 1$.
I... | $$I=
\int _0^{2\pi} \frac{d\theta}{a + \cos\theta}
$$
$$
=2\int _0^{\pi} \frac{d\theta}{a + \cos\theta}
$$
as $\int _0^{2\pi} \frac{d\theta}{a + \cos\theta}=\int _0^{\pi} \frac{d\theta}{a + \cos\theta}+\int _\pi^{2\pi} \frac{d\theta}{a + \cos\theta}$
Now , $\int _\pi^{2\pi} \frac{d\theta}{a + \cos\theta}=-\int ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Integrating $\int \frac{dx}{(x+\sqrt{x^2+1})^{99}}$ I am bugged by this problem: how do I evaluate this?
$$\int \frac{dx}{(x+\sqrt{x^2+1})^{99}}.$$
A closed form will be convenient and fine. Thanks (it does not seem particularly inpiring).
| I like Bitrex's answer best, and my other answer next, but here is one without any substitution. Multiply by $\frac{(\sqrt{x^2+1}-x)^{99}}{(\sqrt{x^2+1}-x)^{99}}$ and you have $$\int\frac{(\sqrt{x^2+1}-x)^{99}}{1}\,dx=\int(\sqrt{x^2+1}-x)^{99}\,dx$$ The next part isn't fun to write out, but you could use the binomial t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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$\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\geq \frac{3}{\sqrt{2}}$ Let $a, b, c$ be positive real numbers such that $a\geq b\geq c$ and $abc=1$
prove that $$\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\geq \frac{3}{\sqrt{2}}$$
| The function $$f(x)=\frac{1}{\sqrt{x}}$$ is convex. Applying Jensen as follows :
$$
a*f[a + b] + b*f[b + c] + c*f[c + a] >= (a + b + c)*f[\frac{
(a (a + b) + b (b + c) + c (c + a))}{(a + b + c)}] = \frac{(a + b + c)^{3/2}}{\sqrt{a^2 + a b + b^2 + a c + b c + c^2}}
$$
We need to prove $$\frac{(a + b + c)^{3/2}}{\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 2
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On the number of possible solutions for a quadratic equation. Solving a quadratic equation will yield two roots:
$$\frac{-\sqrt{b^2-4 a c}-b} {2 a}$$
and:
$$\frac{\sqrt{b^2-4a c}-b}{2 a}$$
And I've been taught to answer it like:
$$\frac{\pm\sqrt{b^2-4a c}- b}{2 a}$$
Why does it yields only two solutions? Aren't there i... | A more important question: Why should there be infinitely many solutions?
The roots/solutions of a quadratic equation $ax^2 + bx + c = 0$ are given by looking at when the graphs $y = ax^2 + bx + c$ cuts the $x$-axis. Remember that $a$, $b$ and $c$ are all fixed numbers, e.g. $y = x^2 + 2x + 1.$ Ask yourself this: Why ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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For some integers $m,n$,proving that $p\mid m$ Assume $p$ is a prime number such that $p\equiv 1 \pmod3$, and $q=\lfloor \frac{2p}{3}\rfloor$.
If:
$$\frac{1}{1\cdot2} +\frac{1}{3\cdot4} +\cdots+\frac{1}{(q-1)\cdot q} =\frac{m}{n}$$
For some integers $m,n$, what is the proof that $p\mid m$
| Let $p=3k+1$ and $H_n$ denote the n-th harmonic number. Since $\displaystyle \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$ (1) the sum in the question is $$\displaystyle \sum_{i=1}^q \frac{ (-1)^{i+1} }{i} = H_{2k}-H_k.$$
Working in mod $p$, we add $p$ to the denominator of each term in $H_k$ : $$ H_{2k} - H_k \equiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Finding all $x$ for $\frac{2x - 13}{2x + 3} \lt \frac{15}{x}$ I'm trying to find all $x$ for the inequality $\frac{2x - 13}{2x + 3} \lt \frac{15}{x}$. In order to do this, I want to factor one side so that I can find all values where $x$ determines the term to equal $0$.
$$\frac{2x - 13}{2x + 3} \lt \frac{15}{x} \iff \... | Hint: From here:
$$\frac{2x^2 - 43x - 45}{2x(x + \frac{3}{2})} \lt 0$$
you could say that the fraction is less than zero if and only if either (1) the numerator is positive and the denominator is negative or (2) the numerator is negative and the denominator is positive. So for case (2), you solve:
$$
2x^2 - 43x - 45 <0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Solving the cubic polynomial equation $x^3+3x^2-5x-4=0$
How can I solve the cubic polynomial equation $$x^3+3x^2-5x-4=0$$
I simplified it to:
$$x(x^2+3x-5)=4$$
But I don't know where to go from here.
| You are given $x^3+3x^2-5x-4=0$
First, find one of the roots of the polynomial. By way of the rational root theorem, we find that -4 is one of the roots, therefore x + 4 is a factor. Next, split the polynomial in accordance with x + 4 as follows:
$x^3 + 4x^2 - x^2 -4x -x - 4 = 0$
Next factor each pair of terms from lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Numerically evaluating the limit of $\frac{x^4-1}{x^3-1}$ as $x\rightarrow 1$ What is the limit as $x \to 1$ of the function
$$ f(x) = \frac{x^4-1}{x^3-1} . $$
| Note that $x^4-1=(x-1)(x^3+x^2+x+1)$ and $x^3-1=(x-1)(x^2+x+1)$.
Thus
$$f(x)=\frac{(x-1)(x^3+x^2+x+1)}{(x-1)(x^2+x+1)}.$$
When $x\ne 1$, the $x-1$ terms cancel. Now we can safely let $x$ approach $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/188607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
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Why is $a^n - b^n$ divisible by $a-b$? I did some mathematical induction problems on divisibility
*
*$9^n$ $-$ $2^n$ is divisible by 7.
*$4^n$ $-$ $1$ is divisible by 3.
*$9^n$ $-$ $4^n$ is divisible by 5.
Can these be generalized as
$a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer?
But why is $a^n$ $-$ $b^n$$ ... | Since you originally observed your pattern while doing proofs by induction, here is a proof by induction on $n$ that $a-b$ divides $a^n - b^n$ for all $n \in \mathbb{N}$:
The statement is clearly true for $n = 1$. Assume the statement is true for $n = m$ for $m \geq 1$. Thus, $a^m - b^m = (a-b)k$, for some $k \in \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 8,
"answer_id": 7
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Inequality $(a+\frac{1}{b})^2+(b+\frac{1}{c})^2+(c+\frac{1}{a})^2\ge 16$ For every real positive number $a,b,c$ such that $ab+bc+ca=1$, how to prove that:
$$\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2\ge 16$$
| By Cauchy-Schwarz inequality,
$$\begin{eqnarray*}
& & \left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2 \\
&=& \sqrt{\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2}\sqrt{\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2+\left(a+\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Find a plane perpendicular to a plane passing by point In $\mathbb R^4$ I have: $$\pi: \begin{cases} x+y-z+q+1=0 \\ 2x+3y+z-3q=0\end{cases}$$
I have to find $\pi' \bot$ $ \pi $ and passing by $P=(0,1,0,1)$. How can I do that? Thanks a lot!
| The equation of any plane$(\pi_1)$ passing through $P(0,1,0,1)$ is $a(x-0)+b(y-1)+c(z-0)+d(q-1)=0$ where $a,b,c,d$ are indeterminate constants,
If $\pi_1 \bot \pi $ , the sum of the product of the directional cosines will be $0$.
So, $(1)(a)+(1)(b)+(-1)(c)+(1)(d)=0\implies a+b-c+d=0$
and $(2)(a)+(3)(b)+(1)(c)+(-3)(d)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/190064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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How to check if a point is inside a rectangle? There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?
| area of the rectangle
Ai: areas of the triangles shown in the pictures. (i = 1, 2, 3, 4)
ai: lengths of the edges shown in the pictures. (i = 1, 2, 3, 4)
bi: lengths of the line segments connecting the point and the corners. (i = 1, 2, 3, 4)
If the point is inside the rectangle, the following equation holds:
$ \mathbf{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/190111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "229",
"answer_count": 24,
"answer_id": 13
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When is a sum of consecutive squares equal to a square? We have the sum of squares of $n$ consecutive positive integers: $$S=(a+1)^2+(a+2)^2+ ... +(a+n)^2$$ Problem was to find the smallest $n$ such, that $S=b^2$ will be square of some positive integer. I found an example for $n=11$. Now, I'm trying to prove, that if $... | Below is a reasonable (but not very illuminating) proof. Put $S_n(x)=\sum_{k=1}^{n} (x+k)^2$. Note that it is also true that for $2<n<11$, $S_n(x)$ is never a square modulo $n^2$, and $S_n(x)$ is also never a square modulo $900$. Perhaps this will inspire others to produce more intelligent proofs.
Here it goes :
$$
\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 3,
"answer_id": 0
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Prove that: $ \int_{0}^{\infty} \frac{2 x \sin x+\cos 2x-1}{2 x^2} = 0$ How would you prove that?
$$ \int_{0}^{\infty} \frac{2 x \sin x+\cos 2x-1}{2 x^2} dx= 0$$
I'm looking for a solution at high school level if possible. Thanks.
| $\cos 2x=1-2\sin^2x$
so, $$\frac{2x\sin x+\cos 2x-1}{2x^2}=\frac{2x\sin x-2\sin^2 x}{2x^2}=\frac{\sin x}{x}-\frac{\sin^2 x}{x^2}$$
so, $$\int_0^{\infty}\frac{2x\sin x+\cos 2x-1}{2x^2}dx=\int_0^{\infty}\frac{\sin x}{x}dx-\int_0^{\infty}\frac{\sin^2 x}{x^2}dx$$
Now, $$\int_0^{\infty}\frac{\sin x}{x}dx=\pi/2$$
and $$\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing that $ \frac{a}{a^2+3}+\frac{b}{b^2+3}\leq\frac{1}{2}$ for $a,b > 0$ and $ab = 1$ using rearrangement inequalities Please help to solve the following inequality using rearrangement inequalities.
Let $a \gt 0$, $b \gt0$ and $ab=1$. Prove that
\begin{equation}\frac{a}{a^2+3}+\frac{b}{b^2+3}\leq\frac{1}{2}.\e... | I don't have a rearrangement inequality proof yet, but I really like the following proof I got.
First note that $a+b \ge 2 \sqrt{ab} = 2$ by AM-GM.
$a^2 + 3 = a^2 + 3ab = a(a+3b) \ge a(2 + 2b) = 2ab(a + 1) = 2(a+1)$, and $b^2 + 3 = b^2 + 3ab = b(b+3a) \ge b(2 + 2a) = 2b(1+a)$.
Thus, we have
$$\frac{a}{a^2+3}+\frac{b}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/191431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
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Compute $ \int_{0}^{1}\frac{\ln(x) \ln^2 (1-x)}{x} dx $ Compute
$$ \int_{0}^{1}\frac{\ln(x) \ln^2 (1-x)}{x} dx $$
I'm looking for some nice proofs at this problem. One idea would be to use Taylor expansion and then integrating term by term. What else can we do? Thanks.
| In this answer I will make use of a Maclaurin series expansion for the term $\ln^2 (1 - x)$, which I show here to be
$$\ln^2 (1 - x) = 2 \sum_{n = 2}^\infty \frac{H_{n - 1} x^n}{n},$$
and the well-known Euler sum of
$$\sum_{n = 1}^\infty \frac{H_n}{n^3} = \frac{1}{2} \zeta^2 (2),$$
several proofs for which can be found... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
An intriguing definite integral: $\int_{\pi/4}^{\pi/2} \frac{x^2+8}{(x^2-16)\sin (x) + 8 x \cos(x)} \ dx$ I need some hints, suggestions for the following integral
$$\int_{\pi/4}^{\pi/2} \frac{x^2+8}{(x^2-16)\sin (x) + 8 x \cos(x)} \ dx$$
Since it's a high school problem, I thought of some variable change, integration ... | Let $R\sin A=x^2-16$ and $R\cos A=8x\implies R=x^2+16$ and $\cos A=\frac{8x}{x^2+16}$
So, $(x^2-16)\sin x+8x\cos x$
$=(x^2+16)(\cos A \cos x+\sin A\sin x)$
$=(x^2+16)\cos (x-A)$
$=(x^2+16)\cos(x-\cos^{-1}(\frac{8x}{x^2+16}))$
Putting $y=x-\cos^{-1}(\frac{8x}{x^2+16})$, we get $\frac{dy}{dx}=\frac{x^2+8}{x^+16}$
$$\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Half Range Sine Series Question:
It is known that $f(x)=(x−4)^2$ for all $x\in [0,4]$.
Compute the half range sine series expansion for $f(x)$.
My answer :
Half range series: $p=8$, $l=4$, $a_0=a_n=0$.
$$b_n=\frac{2}{L}\int_{0}^{L}f(x)\sin\left(\frac{n\pi x}L\right)d(x)=\frac{2}{4}\int_{0}^{4}(x-4)^2\sin\left(\frac{n\p... | The integral defining your $b_n$ is correct, but the final answer is wrong. It differs from mine by ${64\over n\pi}$.
Here's my work from Mathematica:
Here's graphical verification that we are indeed computing the series correctly (I took the 5th, 10th, and 50th partial sum, respectively):
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/193055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proving $\sqrt{a+\sqrt{b}}=\sqrt{m}+\sqrt{n}\iff a^{2}-b$ is a square This is an exercise for the book Abstract Algebra by Dummit and Foote
(pg. 530):
Let $F$ be a field of characteristic $\neq2$ . Let $a,b\in F$ with $b$
not a square in $F$. Prove $\sqrt{a+\sqrt{b}}=\sqrt{m}+\sqrt{n}$ for
some $m,n\in F$ iff $a^{... | Let $F$ be a field of charcteristic different from 2. Let $a$ and $b$ be elements of the field $F$ with $b$ not a square in F. Prove that a necessary and sufficient condition for $\sqrt{a+\sqrt{b}}={\sqrt{m}+\sqrt{n}}$ for $m,n\in F$ is that $a^2-b$ is a square in $F.$
Solution.
$\Rightarrow:$ Suppose that $a^2-b$ is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 0
} |
Give algebraic and geometric descriptions of the $\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}$ Give algebraic and geometric descriptions of $\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}$ where
$a_1 = (1, -1, -2), a_2 = (3, -3, -1), a_3 = (2, -2, -4), a_4 = (2, -2, 1)$
So far, I have:
$$
\begin{matrix}
\;\;\,1 & \;\;\... | $\begin{pmatrix}
1 & 3 & 2 & 2\\
-1& -3& -2& -2\\
-2& -2& 4& 1
\end{pmatrix}$
only use elementary row operation,we can get
$
\begin{pmatrix}
1 &0 &2 &-1 \\
0&1 & 0 &1 \\
0&0 &0 &0
\end{pmatrix}$
then,$a_1=2a_3$,and $a_4=-a_1+a_2$
$\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}=\operatorname{Span} \{ a_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Simple analytic geometry question I need help with Give the equation of a circle with the center $ (a,0) $ which is tangent to the line $ y = x $
I now have $ (x-a)^2 + y^2 = r^2 $ but I don't know how to continue.. please help!
| Let's calculate the intersection of $y=x$ and the circle.
So, $(x-a)^2+x^2=r^2$ as $y=x$,
$\implies 2x^2-2ax+a^2-r^2=0$
As $y=x$ is a tangent of the circle, the roots of the above equation must be same, so that the two points of intersection coincide.
So, $(-2a)^2=4\cdot 2\cdot (a^2-r^2)$ (as the discriminant$(B^2-4AC... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/195594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Limit of the difference quotient of $f(x) = \frac{2}{x^2}$, as $x\rightarrow x_0$ Could someone please show me how to derive the limit of the difference quotient of $f(x) = \frac{2}{x^2}$, as $x\rightarrow x_0$
The difference quotient is just the expression: $(f(x+h)-f(x))/h$
So, it's easy to get an expression for this... | The trick is simply to add the two fractions together.
$$\lim_{h \to 0} \frac{\frac{2}{(x + h)^2} - \frac{2}{x^2}}{h} = \lim_{h \to 0} \frac{\frac{2[x^2 - (x+h)^2]}{x^2(x + h)^2}}{h} = \lim_{h \to 0} \frac{\frac{2[-2hx - h^2]}{x^2(x + h)^2}}{h} = \lim_{h \to 0} \frac{2[-2x - h]}{x^2(x + h)^2} = \frac{2(-2x)}{x^4} = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/195785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
how to calculate the exact value of $\tan \frac{\pi}{10}$ I have an extra homework: to calculate the exact value of $ \tan \frac{\pi}{10}$.
From WolframAlpha calculator I know that it's $\sqrt{1-\frac{2}{\sqrt{5}}} $, but i have no idea how to calculate that.
Thank you in advance,
Greg
| Let $\theta=\frac\pi{10}$ and $\tan\theta=x$. Then $5\theta=\frac\pi2$ so
\begin{align}
\tan4\theta&=\frac1x\tag{1}
\end{align}
By twice using the double-angle tan formula,
\begin{align*}
\tan2\theta&=\frac{2x}{1-x^2}\\
\tan4\theta&=\frac{2(\frac{2x}{1-x^2})}{1-(\frac{2x}{1-x^2})^2}\\
\frac1x&=\frac{4x(1-x^2)}{(1-x^2)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/196067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 5
} |
Maclaurin expansion of $\arcsin x$ I'm trying to find the first five terms of the Maclaurin expansion of $\arcsin x$, possibly using the fact that
$$\arcsin x = \int_0^x \frac{dt}{(1-t^2)^{1/2}}.$$
I can only see that I can interchange differentiation and integration but not sure how to go about this. Thanks!
| If I was doing this I would start with one of the very common series like $\sin(x)$, $e^x$, etc. and then use substitution. The one that fits best here in my opinion is:
$$
\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \frac{5x^4}{128} + \dotsc
$$
Through substitution, we can obtain:
$$
\frac{1}{\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/197874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 8,
"answer_id": 5
} |
Perfect squares Wonder whether anybody here can provide me with a hint for this one.
Is $c=1$ the only case in which the expression
$(c^2+c-1)(c^2-3(c-1))$
returns a perfect square?
| Yes, $c=0$ is the only such value. For the proof, it is useful to let $c=x+1$. Then our expression becomes
$$(x^2+3x+1)(x^2-x+1).$$
Note that $x^2-x+1$ is always odd. Any common divisor of $x^2+3x+1$ and $x^2-x+1$ must divide the difference $4x$. But such a common divisor must be odd, so any common divisor must divide... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/198129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the inverse a matrix with trigonometic entries What is the inverse of
\[
\begin{pmatrix}
1&0&0\\0&\cos x &\sin x\\ 0 &\sin x &-\cos x \end{pmatrix}
\]
Please help me to solve the above problem.
| Let $A_x :=\left[ \begin{array}{cc} \cos x & \sin x \\ \sin x & -\cos x \end{array} \right]$, and let $R_x :=\left[ \begin{array}{cc} \cos x & -\sin x \\ \sin x & \cos x \end{array} \right]$ be the matrix of the rotation by angle $x$ in the plane (that is, for all ${\bf v}$ in $\mathbb R^2$, $\ R_x\cdot {\bf v}$ is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Determine the number of solutions of the equation $n^m = m^n$
Possible Duplicate:
$x^y = y^x$ for integers $x$ and $y$
Determine the number of solutions of the equation
$n^m = m^n$
where both m and n are integers.
| Hint:
Since $m^n=n^m$, take logs and separate the variables:
$$
\frac{\log(m)}{m}=\frac{\log(n)}{n}
$$
This suggests considering the function $f(x)=\frac{\log(x)}{x}$.
$\hspace{2cm}$
Another Approach:
Start by comparing $n^{n+1}$ vs $(n+1)^n$. Divide both by $n^n$, to get $n$ vs $\left(1+\frac1n\right)^n$. We can use t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
} |
finding $\frac{\sin 2x}{\sin 2y}+\frac{\cos 2x}{\cos 2y}$ If:
$$\frac{\cos x}{\cos y}=\frac{1}{2}$$ and $$\frac{\sin x}{\sin y}=3$$
How to find
$$\frac{\sin 2x}{\sin 2y}+\frac{\cos 2x}{\cos 2y}$$
| Let $$\frac {\cos x}{1}=\frac {\cos y}{2}=a(say),\implies \cos x=a,\cos y =2a$$
and $$\frac{\sin x }{3}=\frac{\sin y }{1}=b(say),\implies \sin x=3b, \sin y =b$$
So, $$a^2+(3b)^2=1,(2a)^2+b^2=1\implies a^2=\frac 8{35}, b^2=\frac 3{35}$$
$$\implies \cos^2x=a^2=\frac 8{35},\sin^2y=b^2=\frac 3{35}$$
So, $$\frac{\sin 2x}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Regd. Riemann Rearrangement Assume that A is an arbitrary set and there exists a bijection $\phi : B \rightarrow A$ and $x_{\alpha} \in [0,\infty]$, the book says that
$$\Sigma_{\alpha \in A} x_{\alpha}= \Sigma_{\beta\in B} x_{\phi(\beta)}$$
can fail if the series is not absolutely convergent and we are dealing with ... | If a series is not absolutely convergent, then it only converges because some positive and negative terms cancel in the right way as you continue summing. However, upon rearrangement, you may find that the positive and negative terms do not cancel the same way. Consider the series
$$ 1 - \frac{1}{2} - \frac{1}{2} + \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/203794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Solve the inequality $|z^2|-|z|\ \Re(z)>0$ Determine the set of complex numbers $z$ such that
$|z|^2-|z|\ \Re(z)>0$
This is my process:
Putting $z=x+iy$, we have $\Re(z)=x$ (real part), $|z^2|=x^2+y^2$, $|z|=\sqrt{x^2+y^2}$, and:
$(x^2+y^2)-x\sqrt{x^2+y^2}>0\Rightarrow x^2\left(1+\frac{y^2}{x^2}\right)-x|x|\sqrt{1+\fra... | I think from $$(x^2+y^2)-x\sqrt{x^2+y^2}>0$$
$$\to (x^2+y^2) > x\sqrt{x^2+y^2}$$
If $x$ and $y$ are not both $0$, divide by $\sqrt{x^2+y^2}$ on both side
$$\sqrt{x^2+y^2} > x$$
which is always true as long as $y\neq 0$ or $x< 0$.
So the solution would be $z=\{x+iy: y\neq 0\ \text{ or } x <0\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/207106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Derivative of $\frac{1}{\sqrt{x+5}}$
I'm trying to find the derivative of $\dfrac{1}{\sqrt{x+5}}$
using $\displaystyle \lim_{h\to 0} \frac {f(x+h)-f(x)}{h}$
So,
$$\begin{align*}
\lim_{h\to 0} \frac{\dfrac{1}{\sqrt{x+h+5}}-\dfrac{1}{\sqrt{x+5}}}{h} &= \frac{\dfrac{\sqrt{x+5}-\sqrt{x+h+5}}{(\sqrt{x+h+5})(\sqrt{x+5})... | Do you know the derivative of logarithms?
$$
f(x)=\frac{1}{\sqrt{x+5}}\\
Lf(x)=\log f(x)=-\frac{1}{2}\log(x+5)\\
\frac{f'(x)}{f(x)}=\bigg( -\frac{1}{2}\log(x+5) \bigg)'_x\\
f'(x)=f(x)\bigg( -\frac{1}{2}\log(x+5) \bigg)'_x\\
L_1=\lim_{h \to 0}\frac{-\frac{1}{2}\log(x+5+h)+\frac{1}{2}\log(x+5)}{h}=-\frac{1}{2}\lim_{h \to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/207484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to solve the cubic equation $x^3-12x+16=0$ Please help me for solving this equation $x^3-12x+16=0$
| Hint: $x=2$ is a solution of the polynomial.
So $x^3-12x+16$ is divisible by $(x-2)$.
And $x^3-12x+16= (x-2)(x^2+2x-8)$.
The roots of $x^3-12x+16=0$ are $x=2$ and the roots of $x^2+2x-8=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/208183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Show $\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$ How to show the following equality?
$$\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$$
| This is what I have from an essay I wrote. I don't know if there's a more elementary way (or if it's completely correct).
Consider $f(z) = \dfrac{\cot{\pi z}}{z^2 + k}$. This will have residues at $z = \pm i \sqrt{k}$, and at $z = n$ for $n \in \mathbb{Z}$. At $z = n$, we can compute the residues as
\begin{align*} \tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "44",
"answer_count": 5,
"answer_id": 1
} |
Solve $5a^2 - 4ab - b^2 + 9 = 0$, $ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0$
Solve $\left\{\begin{matrix} 5a^2 - 4ab - b^2 + 9 = 0\\ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0. \end{matrix}\right.$
I know that we can use quadratic equation twice, but then we'll get some very complicated steps. Are there any elegant wa... | Note that
\begin{equation*}
4(5a^2 - 4ab - b^2 + 9) - 9(-21a^2-10ab+40a-b^2+8b-12) = (19a+5b-12)(11a+b-12).
\end{equation*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/210454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
} |
Findind the value of $x^4 + 1/x^4$ when $x = 2+3\sqrt{3}$ This question recently came up in a competitive exam and I have been struggling to find out the easiest way to solve it.
Given that $x = 2+3\sqrt{3}$, what is the value of $x^4 + 1/x^4$.
Could someone provide me with an approach other than the direct approach of... | You the thing you need to calculate is $x^4$.
You start by isolating the square root $x = 2+3\sqrt{3} \Leftrightarrow x - 2 = 3\sqrt{3}$
Then you square both sides $(x - 2)^2 = (3\sqrt{3})^2 \Leftrightarrow x^2 - 4x + 4 = 27 \Leftrightarrow x^2 - 4x -23 = 0$
Let $P$ be $X^4$
Let $D$ be $X^2 - 4X - 23$
Now, you compute ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/212075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Proving trigonometric Identity: $\frac{1+\sin x}{\cos x} = \frac{1+\sin x+\cos x}{1-\sin x+\cos x}$ I would like to try and prove
$$\frac{1+\sin x}{\cos x} = \frac{1+\sin x+\cos x}{1-\sin x+\cos x}$$
using $LHS=RHS$ methods, i.e. pick a side and rewrite it to make it identical to the other side.
I found a quick way by ... | For $b\not=0$ we have:
$$\begin{align*}\frac{1+a}{b} &= \frac{1+a+b}{1-a+b}\qquad&\iff \\
(1+a)(1-a+b) &= b(1+a+b)\qquad&\iff\\
1-a^2+b(a+1) &= b^2 + b(a+1)\qquad&\iff\\
a^2+b^2&=1
\end{align*}$$
So your equation is an alternate way to characterize $\cos^2 x+\sin^2 x = 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/213788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
} |
find a point on ellipse closest to origin Find the points on the ellipse $2x^2 + 4xy + 5y^2 = 30$ closest and farthest from origin. How to do this problem? I know how to find a closest point if $z = f(x,y)$ is given, however, this is 2 dimensional.
| Another way is use Rotation of axes, to eliminate the $xy$ term.
Here $\cot 2\theta=\frac{2-5}4=-\frac 3 4$
$\frac {\cos 2\theta}3=\frac{\sin 2\theta}{-4}=\frac 1{\pm 5}$ (Using squaring & adding)
If $\cos 2\theta=\frac 3 5,\sin 2\theta=-\frac 4 5$
Using $\cos 2\theta=2\cos^2\theta-1=1-2\sin^2\theta$, we get $\sin^2\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
} |
Mechanism Behind Dot Product and Least Square Sorry for my ignorance, but I want to know how the mechanism of finding the least square solutions or the closest points in Euclidean space works.
For example:
Find the closest point or points to $b =(−1,2)^T$ that lie on the line $x + y = 0$.
I know the answer is
$$\f... | The projection onto a vector $x$ is give by $$Pv=\frac{xx^T}{||x||^2}v,$$ where $P \equiv \frac{xx^T}{||x||^2}$ is the projection operator, and $||x||^2=(x,x)$ (or $x*x$ in your notation) is the normalization factor.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/214577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $\dfrac{4x^2-1}{4x^2-y^2}$ is an integer, then it is $1$ The problem is the following:
If $x$ and $y$ are integers such that $\dfrac{4x^2-1}{4x^2-y^2}=k$ is
also an integer, does it implies that $k=1$?
This equation is equivalent to $ky^2+(1-k)4x^2=1$ or to $(k-1)4x^2-ky^2=-1$. The first equation is a pell equat... | Well notice that both the numerator and denominator are squares, so we may factor
\begin{equation}
\frac{4x^2 - 1}{4x^2 - y^2} = \frac{(2x+1)(2x-1)}{(2x-y)(2x+y)}.
\end{equation}
It is easy to see that $\gcd(2x + 1, 2x - 1) = 1$, so if $k$ is an integer, it must be the case that the denominator each divides one of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
} |
Number Theory: Prime modulus and Wolstenholme's congruence I apologize if my title is inappropriate, but I couldn't think of a better title name pertaining to this particular problem I have other than listing the section's name of this particular textbook I am using. I've been working on this number theory for a long t... | $$2^p=(1+1)^p=\sum_{k=0}^{p}\binom{p}{k}=2+\sum_{k=1}^{p-1}\binom{p}{k}.$$
Now $\binom{p}{k}/p=\frac{(p-1)\dots(p-(k-1))}{1\dots k}\equiv(-1)^{k-1}1/k$ mod $p$ (for $1\leq k\leq p-1$), so we are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/218894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Let$\frac{\text{dy}}{\text{d}x}=\frac{3y+1}{x^2}$ ,What is $y(x)$? Let$$\frac{\text{dy}}{\text{d}x}=\frac{3y+1}{x^2}$$
What is $y(x)$?
I tried anti-differentiation,but it seems does not work. Is there any tricks to solve the problem?
| $$\begin{align*}\frac{1}{3y+1}dy&=x^{-2}dx\\
\frac{1}{3}\int \frac{3}{3y+1} dy&=\int x^{-2}dx\\
\frac{1}{3}\ln|3y+1|&=-x^{-1}+C\\
3y+1&=Ae^{-\frac{3}{x}}\text{, where }\ln A=3C\\
y&=\frac{1}{3}(Ae^{-\frac{3}{x}}-1)\\
y&=Be^{-\frac{3}{x}}-\frac{1}{3}\text{, where }B=\frac{1}{3}A
\end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/220802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Differentiating $x^2 \sqrt{2x+5}-6$ How do I differentiate this function: f(x)= $x^2 \sqrt{2x+5}-6$
I had: I had $2x\sqrt{2x+5} + x^2 \dfrac{1}{2\sqrt{2x+5}}$ but the correction model said it was I had $2x\sqrt{2x+5} + x^2 \dfrac{2}{2\sqrt{2x+5}}$
| $$
\begin{eqnarray*}
y &=& x^2(2x+5)^{1/2} - 6 \\
\frac{dy}{dx} &=& \frac{d}{dx} \Big[ x^2(2x+5)^{1/2} \Big] - \frac{d}{dx}\Big[ 6 \Big], \qquad \textrm{Sum/Difference Rule}\\
&=& \frac{d}{dx}\Big[x^2\Big](2x+5)^{1/2} + x^2\frac{d}{dx}\Big[(2x+5)^{1/2}\Big] - 0, \qquad \textrm{Product Rule}\\
&=& 2x(2x+5)^{1/2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/220858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
For what value of m that the equation $y^2 = x^3 + m$ has no integral solutions? For what value of m does equation $y^2 = x^3 + m$ has no integral solutions?
| None of the solutions posted look right (I don't think this problem admits a solution by just looking modulo some integer, but possibly I'm wrong). Here is a proof.
First, by looking modulo $8$ one deduces we need $x$ to be odd.
Note that $y^2 + 1^2 = (x+2)(x^2 - 2x + 4)$. As the LHS is a sum of two squares, no primes ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Numerical method for finding the square-root. I found a picture of Evan O'Dorney's winning project that gained him first place in the Intel Science talent search. He proposed a numerical method to find the square root, that gained him $100,000 USD.
Below are some links of pictures of the poster displaying the method.
... | The iteration to find $\sqrt k$ is
$f(x) = \frac{d x+k}{x+d}$
where $d = \lfloor \sqrt k \rfloor$.
The iterations start with $x = d$.
If $x$ is a fixed point of this,
$x = \frac{d x+k}{x+d}$,
or $x(x+d) = dx + k$
or $x^2 = k$,
so any fixed point must be the square root.
Now wee see if the iteration increases or decreas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Inequality. $\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a} \geq \frac{ab+bc+ca}{2}$ prove the following inequality:
$$\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a} \geq \frac{ab+bc+ca}{2},$$ for $a,b,c$ real positive numbers.
Thanks :)
| By the Cauchy-Schwarz Inequality: $$\frac{a^4}{a^2+ab} + \frac{b^4}{b^2 + bc} + \frac{c^4}{c^2 + ac} \ge \frac{(a^2+b^2+c^2)^2}{a^2 + b^2 + c^2 + ab + ac + bc}$$
Now, I claim that $$\frac{(a^2+b^2+c^2)^2}{a^2 + b^2 + c^2 + ab + ac + bc} \ge \frac{ab+ac+bc}{2}$$
Expanding, it suffices to show: $$2a^4 + 2b^4 + 2c^4 + 4a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
} |
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.