Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Factor $1+x+x^2+x^3+...+x^{14}$ In a previous task, I was asked to factor $1+x+x^2+x^3$ for $x \in \mathbb{R}$, which I accomplished by solving
$1+x+x^2+x^3 = 0 \to $
$1+x(1+x+x^2) = 0 \to $
$x(1+x+x^2) = -1$
which has a solution $x = -1$, and thus I knew $(x+1)$ was a factor. A bit of guesswork gave me $(x+1)(x^2+1)$... | Careful about concluding from $x(1+x+x^2+...+x^{13}) = -1$ that $x=-1$ must be a solution:
$$\textrm{When}\;\; x = -1,\;\; x(1+x+x^2+...+x^{13}) = -1 [7(1) + 7(-1)] = -1(0) = 0\neq -1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/249988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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Finding the minimum of $\frac pq + \frac rs$ for distinct integers $p, q, r, s$ from $\{1,2,3,4,5,\ldots,16,17\}$ Here is the question:
Four distinct integers $p$, $q$, $r$ and $s$ are chosen from the set $\{1, 2, 3, 4, 5, \ldots, 16, 17\}$. The minimum possible value of $\frac pq + \frac rs$ can be written as $\frac ... | $\frac{1}{16} + \frac{2}{17} = \frac{1}{16} + \frac{1}{17} + \frac{1}{17}$
$\frac{2}{16} + \frac{1}{17} = \frac{1}{16} + \frac{1}{16} + \frac{1}{17}$
So obviously the first one is smaller.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/250607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Calculate the sum of series $\sum\limits_{i=0}^{n-1} i2^i$
Possible Duplicate:
How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$?
How can I calculate precise value of that series: $\sum\limits_{i=0}^{n-1} i2^i$ ?
So far, I tried to differentiate the $ \sum\limits_{i=0}^{n} 2^i = 2^{i-1} - 1 $ series, b... | $$x+x^2+x^3+...+x^{n-1}+x^n=\frac {x^{n+1}-x}{x-1}$$
$+$
$$0x+x^2+x^3+...+x^{n-1}+x^n=\frac {x^{n+1}-x^2}{x-1}$$
$+$
$$0x+0x^2+x^3+...+x^{n-1}+x^n=\frac {x^{n+1}-x^3}{x-1}$$
$$.$$
$$.$$
$$.$$
$+$
$$0x+0x^2+0x^3+...+0x^{n-1}+x^n=\frac {x^{n+1}-x^n}{x-1}$$
After adding we get:
$$x+2x^2+...+nx^n=\sum_{i=1}^{n}\frac {x^{n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/250746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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How come $2 \times 3^k + 3^k = 3 \times 3^k$ I get something else
$$\begin{align*}
&2 \times 3^k + 3^k=\\
&2 \times 3^k \times 2=\\
&4 \times 3^k
\end{align*}$$
What does $3^k + 3^k$ give exactly?
| It looks like you are confusing $2\cdot 3^k+3^k$ with $2 \cdot (3^k+3^k)$. The first gives $3 \cdot 3^k=3^{k+1}$, while the second gives $2 \cdot (2 \cdot 3^k)=4\cdot 3^k$. Parentheses are important.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/251905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
} |
How to integrate $(x^{1/2} + 5 x^{1/3})^{-1}$? $$ \int_0^1 \frac{dx}{x^\frac12+5x^\frac13} $$
Let $$u = 5+x^\frac16 $$
I have change x into u, and find $$\int_5^6 \frac{7}{6u(u-5)^3} $$
Then, i don't know how to integrate it.
| $$
\int \frac{1}{u(u-5)^3}du=\int\frac{A}{u}+\frac{B}{(u-5)}+\frac{C}{(u-5)^2}+\frac{D}{(u-5)^3}du.
$$
Courtesy of partial fraction decomposition. :-)
Edit: It would appear your $u$ sub does not work. (Thanks, Gerry.) Nonetheless, I hope this is a helpful illustration of partial fraction decomposition.
Here is why you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/253014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove that $5/2 < e < 3$?
Prove that $$\dfrac{5}{2} < e < 3$$
By the definition of $\log$ and $\exp$,
$$1 = \log(e) = \int_1^e \dfrac{1}{t} dt$$
where $e = \exp(1)$.
So given that $e$ is unknown, how could I prove this problem? I think I'm missing some important facts that could somehow help me rewrite $e$ in s... | Method 1:
Using the power series for $\log(1+x)$, we get
$$
\begin{align}
\log\left(1+\frac1n\right)
&=-\log\left(1-\frac1{n+1}\right)\\
&=\frac1{n+1}+\frac1{2(n+1)^2}+\frac1{3(n+1)^3}+\dots\tag{1}
\end{align}
$$
Multiply $(1)$ by $n+1$:
$$
(n+1)\log\left(1+\frac1n\right)=1+\frac1{2(n+1)}+\frac1{3(n+1)^2}+\dots\tag{2}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 9,
"answer_id": 3
} |
Proof of $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$ I've tried following this way, but I haven't succeeded.
Thank you!
| Calculate the derivative of both sides:
$$(\arctan x)'=\frac{1}{1+x^2}$$
$$\left(\arcsin\frac{x}{\sqrt{1+x^2}}\right)'=\frac{\sqrt{1+x^2}-\frac{x^2}{\sqrt{1+x^2}}}{1+x^2}\cdot\frac{1}{\sqrt{1-\frac{x^2}{1+x^2}}}=$$
$$=\frac{1}{(1+x^2)\sqrt{1+x^2}}\cdot\frac{\sqrt{1+x^2}}{\sqrt 1}=\frac{1}{1+x^2}$$
Since both derivative... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 0
} |
simplify summation of factorial (random walk) I suspect that the expression
$$\sum_{n=0}^N \frac{(N-2n)^2}{n!(N-n)!}$$
simplifies to
$$\frac{2^N}{(N-1)!}$$
But I cannot find the intermediate steps. Can someone give me a hint how I can deduce this result?
(The expression comes up when calculating the average final posi... | $$(N-2n)^2 = (N-n)^2 + n^2 - 2n(N-n)$$
Hence,
\begin{align}
S_N(n) & = \dfrac{(N-2n)^2}{n!(N-n)!}\\
& = \dfrac{(N-n)^2 + n^2 - 2n(N-n)}{n!(N-n)!}\\
& = \dfrac{N-n}{n!(N-n-1)!} + \dfrac{n}{(n-1)! (N-n)!} - 2 \dfrac1{(n-1)!(N-n-1)!}\\
& = \dfrac1{n!(N-n-2)!} + \dfrac1{n!(N-n-1)!} + \dfrac1{(n-2)!(N-n)!} + \dfrac1{(n-1)!(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/258882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 1
} |
No natural numbers sucht that $x+3y+5z=2012$ and $x^2+y^2+3z^2=2013$. Prove that there are not exist the natural numbers, $x,y,z$ such that $x+3y+5z=2012$ and $x^2+y^2+3z^2=2013$.
tell me please if my proof is ok ?
$x^2+y^2=3\cdot671-3z^2=3(671-z^2)=3k$ but if we want we can find out $z$. $671 \geq z^2$ so $z=\overl... | Just work $\bmod\ 2$: Your first equation reads $x+y+z=0$ and your second reads $x^2+y^2+z^2=1$, but since $x^2\equiv x\bmod\ 2$, the second is equivalent to $x+y+z=1$ and the two are immediately incompatible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/258962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Do there exist any odd prime powers that can be represented as $n^4+4^n$? Well, I wrote up a solution on it, but according to the place I found the problem, it isn't quite correct. Ah, I'm simply hoping someone will point out where I got wrong.
Now, let, $n^4+4^n = p^k$, where $p$ is an odd prime and $k$ is a positive ... | For odd $n>1$ the expression is never prime, it can be factored into a product where each factor is the sum of two squares:
$$[((2^m)+n)^2 +(2^m)^2][(2^m)^2 +((2^m)^2-n)^2],$$
where $m = (n-1)/2$.
Looks like I didn't read the question properly thanks for the feedback.
I will try and improve my answer.
Let the first fa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/261925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
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Geodesic of helicoid What is the geodesic of the helicoid?
$M=\{ (x,y,z)\in \mathbb{R}^3: x\sin z - y\cos z =0\}$
Whose tangent at the point $p = (1,0,0)$ in the line $r = \{(x, y, z) \in \mathbb {R} ^ 3: x = 1, y = z\}$.
I tried with the definition, but I don't know the parameterization, and so do not know which is th... | The general parametrisation is $X(u,v) = \left( u \cos(v) , u \sin(v) , kv \right)$, where $k$ is a non-zero constant (greater than zero for right, or less than for left), and looking at your definition of $M$ it appears that $k=1$. It is always important to remember that there is usually more than one way to parametri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Analytical form of an integral I'm trying to find the analytical form of the following integral:
$$\int_0^{2\pi}\left|\frac{(1-i\cos(\phi))}{(1+a-i\cos(\phi))}\right|^2d\phi$$
I've tried in Wolfram Alpha, which ran out of free calculation time, and in Mathematica I get the original expression for the indefinite integra... | $$
\begin{align}
\int_0^{2\pi}\left|\frac{(1-i\cos(\phi))}{(1+a-i\cos(\phi))}\right|^2\,\mathrm{d}\phi
&=\int_0^{2\pi}\frac{1+\cos^2(\phi)}{(1+a)^2+\cos^2(\phi)}\,\mathrm{d}\phi\\
&=2\pi-\int_0^{2\pi}\frac{(1+a)^2-1}{(1+a)^2+\cos^2(\phi)}\,\mathrm{d}\phi\\
&=2\pi-2\int_0^\pi\frac{(1+a)^2-1}{2(1+a)^2+1+\cos(2\phi)}\,\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that an integral is positive For any positive integer $n$, consider
$$\int_{0}^\infty\frac{(r^2-1)r^{n+1}}{(r^2+1)^{n+3}}dr.$$
I would like to show that it is positive. I try to write it as
$$\int_{0}^\infty\frac{(r^2-1)r^{n+1}}{(r^2+1)^{n+3}}dr=\int_{0}^\infty\frac{r^{n+1}}{(r^2+1)^{n+2}}dr-2\int_{0}^\infty\fr... | If $$\int \frac{(r^2-1)r^{n+1}}{(r^2+1)^{n+3}}dr=I$$
$$\int \frac{(r^2-1)r^{n+1}}{(r^2+1)^{n+3}}dr=\int \frac{r^{n+1}}{(r^2+1)^{n+2}}dr-2\int_{0}^\infty\frac{r^{n+1}}{(r^2+1)^{n+3}}dr$$
Now $$\int\frac{r^{n+1}}{(r^2+1)^{n+2}}dr=\frac1{(r^2+1)^{n+2}}\int r^{n+1} dr-\int\left(\frac{d\frac1{(r^2+1)^{n+2}}}{dr}\int r^{n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Confusion on Eigenvalues of Similar Matrices
Please help me to identify where I went wrong:
The completely reduced normal form of the real matrix $A=
$\begin{pmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1 \\
\end{pmatrix}
is the following matrix $B=
$\begin{pmatrix}
... | Your reduced normal form for the matrix A is wrong.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/263595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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What is the difference between $\lfloor f \rfloor (x)$ and $\lfloor f(x) \rfloor $? I have a very basic question: What is the difference between $\lfloor f \rfloor (x)$ and $\lfloor{f(x)}\rfloor$? Are $\lfloor f \rfloor (x)$ and $f(\lfloor x \rfloor)$ equivalent?
| You seem to be asking two different questions. $\lfloor f \rfloor (x)$ is defined as $\lfloor f(x) \rfloor$. Hence, it is indeed true that $\lfloor f \rfloor (x) = \lfloor f(x) \rfloor$.
However, $\lfloor f(x) \rfloor$ is different from $f(\lfloor x \rfloor)$. For instance, consider $f(x) = x^2$. Then $\lfloor f(x) \rf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Showing that: $(\frac{a}{b+c})^2+(\frac{b}{a+c})^2+(\frac{c}{a+b})^2+\frac{10abc}{(a+b)(b+c)(c+a)}\ge 2$ Let a;b;c>0. Prove:
$$\left(\frac{a}{b+c}\right)^2+\left(\frac{b}{a+c}\right)^2+\left(\frac{c}{a+b}\right)^2+\frac{10abc}{(a+b)(b+c)(c+a)}\geq 2$$
I think
$$\frac{2a}{b+c}=x;\frac{2b}{c+a}=y;\frac{2c}{a+b}=z$$
We ha... | A full expanding gives
$$\sum_{cyc}(a^6+2a^5b+2a^5c-a^4b^2-a^4c^2-4a^3b^3+a^2b^2c^2)\geq0$$ or
$$\sum_{cyc}(a^6-a^4b^2-a^4c^2+a^2b^2c^2)+2\sum_{cyc}ab(a^2-b^2)^2\geq0,$$
which is true by Schur.
Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Finding Eccentricity from the rotating ellipse formula I see that from a normal ellipse formula, we can acquire the eccentricity via this formula here.
However, for this formula (1):
$A(x − h)^2 + B(x − h)(y − k) + C(y − k)^2 = 1$
When parameter $B = 0$, we would have normal ellipse, and the formula from the link abo... | As the translation does not change the eccentricity of a conic, let's us define $x=X+h,y=Y+k$ to get $AX^2+BXY+CY^2=1$
Using Rotation of axes as eccentricity is one of the invariants in rotation,
$$x'^2(A\cos^2t+B\sin t\cos t+C\sin^2t)+x'y'(B\cos2t-(A-C)\sin2t)+y'^2(A\sin^2t-B\sin t\cos t+C\cos^2t)=1$$
To remove the $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
prove that there are no number such that $\overline{ab\ldots}=a^2+b^2+\ldots$ Prove that a natural number with at least 2 digits cannot be written like a sum with the the power of digits equal $2$.
What I want to say: $$\overline{ab}\neq a^2+b^2.$$
What I have done:
$$10a+b=a^2+b^2 $$ or
$$a(a-10)=b(1-b).$$
$b(1-b)=... | For more than 3 digits, if there are $n$ digits the sum of squares is at most $81n$ while the number is at least $10^{n-1}$ and you are done. For specifically $3$ digits, the maximum sum of squares would be $3 \cdot 81=243$, but then the leading digit is no more than $2$, so the maximum sum is no more than $4+162=166$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Prove that $\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{c^2+1} \leq \frac{9}{10}$ if $a+b+c=1$. Let $a,b,c$ are real number such that $a+b+c=1$. Prove that:
$$\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{c^2+1} \leq \frac{9}{10}.$$
| As you have known, the inequality holds when $a,b,c\ge -\frac{3}{4}$. To complete the proof, without loss of generality, we may assume that $a\le -\frac{3}{4}$.
Note that $\frac{b}{b^2+1},\frac{c}{c^2+1}\le \frac{1}{2}$. Since when $-9\le a\le -\frac{3}{4}$, $\frac{a}{a^2+1}<-\frac{1}{10}$, the inequality holds when $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Find the limit of $(x_n)$ defined by $x_{n+1}=c_nu(x_n)$ Find $\displaystyle \lim_{n\rightarrow \infty }x_n$ :
$$\left\{\begin{matrix}x_1=a>0\\ \\ x_{n+1}=\frac{2x_n\cdot \cos\left(\frac{\pi}{2^n+1}\right)}{x_n+1}\end{matrix}\right.$$
I have tried that :
Let $a_n=\dfrac{1}{x_n}$ . So :
$$a_{n+1}=\frac{1}{2\cos\left(\fr... | An incomplete answer that assumes the existence of the limit:
Since $\cos\frac{\pi}{2^n+1}$ goes to one for $n\to\infty$, the limit $x_\infty=\lim_{n\to\infty}x_n$ will, if it exists, satisfy the fixed-point equation $x_\infty=2x_\infty/(x_\infty+1)$. The only solutions of this equation are $0$ and $1$.
The derivative ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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For which values of $x,y$ does $x^2+y^2$ take the minimum?
Find the values of $x,y$ for which $x^2+y^2$ takes the minimum value where $(x+5)^2+(y-12)^2=14^2$.
Trial:$\begin{align} \ (x+5)^2+(y-12)^2=14^2 \\ \implies x^2+y^2+10x-24y=27 \end{align}$.
Here I am stuck. Please help.
| What about Lagrange's Multipliers? Let
$$f(x,y)=x^2+y^2\,\,,\,\,g(x,y)=(x+5)^2+(y-12)^2-14^2\Longrightarrow$$
$$H(x,y,\lambda)=f+\lambda g=x^2+\lambda(x+5)^2+y^2+\lambda(y-12)^2-14^2\Longrightarrow$$
$$H'_x=2\left((x+\lambda(x+5)\right)=0\Longrightarrow x=-\frac{5\lambda}{1+\lambda}$$
$$H'_y=2\left(y+\lambda(y-12)\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/268598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Solve $xy=3$ and $4^{x^2}+2^{y^2}=72$ I have a system of equations $xy=3$ and $4^{x^2}+2^{y^2}=72$ whose solution I know is $x=y=\sqrt 3$, but what are the steps to solve it?
| The first attempt to try to solve a system of two (or more variables) is naturally assumed that a possible solution is one in which all variables are equal.
Then set $x=y=t$ and solve de equation
\begin{cases}
xy=t\cdot t=3\\
4^{x^2}+2^{y^2}=\big(2^{t^2}\big)^2+2^{t^2}=72
\end{cases}
By $t^2=3$ we have $t=\pm\sqrt{3}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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proving that $ab$ is a perfect square. let $a,b \ge 2$ be integers, such that for every positive integer $n$, the expression :
$(a^n-1)(b^n-1)$ is a perfect square.
How to prove that $ab$ is a perfect square.
| It is obvious that for $a=b$ the product $(a^n-1)(b^n-1)$ is square for all $n$. We prove that this is the only possibility.
Let $b^n=a^n+h_n$; one has $(a^n-1)(a^n-1+h_n)=x_n^2$ so
$$(a^n-1)^2+h_n(a^n-1)-x_n^2=0\Rightarrow 2(a^n-1)=-h_n\pm\sqrt{h_n^2+4x_n^2}$$ so we have a Pythagorean triple
$$(h_n, 2x_n,z_n)=(r_n^2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 2
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Euclidean norm and matrix questions Let $A$ be an $n×m$ real matrix and define $\vert A \vert_{2}^{2}={\rm tr}(A^tA)$.
1)Show that $|A|_2$ is the Euclidean norm of $A$, when we view $A$ as a vector in $R^{nm}$ by stacking the columns of $A$.
2)Find the cosine of the angle between
$A=\begin{bmatrix}
1 & 1 & 1\\
1& 0 &... | This: if
$$
A =
\begin{pmatrix}
1 & 3 & 5 \\
2 & 4 & 6
\end{pmatrix}
$$
then, as a vector of $\mathbb{R}^{3\cdot 2}$, we view it like
$$
\begin{pmatrix}
1 \\
2 \\
3 \\
4 \\
5 \\
6
\end{pmatrix}
$$
and the square of its norm is
$$
\vert A \vert_2^2 =
\mathrm{tr}
\left[
\begin{pmatrix}
1 & 2 \\
3 & 4 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
a question about uniform convergence of a series The problem is to investigate the uniform convergence of $\sum\limits_{n=0}^\infty \frac{x^2}{x^2+n^2}.$ Any help will be appreciated.
Thanks in advance.
| As a bonus, this series not only converges but there is a closed form expression for the sum. Using the method presented here, we study the integral of $$ f(z) = \frac{x^2}{x^2+z^2} \pi \cot(\pi z)$$ along a circle of radius $R$ in the complex plane where $R$ goes to infinity. This integral vanishes in the limit, so th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/277289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Is there any general formula for $S = 1^1 + 2^2 + 3^3 + \dotsb+(n - 1)^{n - 1} + n^n, n \in N$? Is there any general formula to sum following series:
$$S = 1^1 + 2^2 + 3^3 + \dotsb+(n - 1)^{n - 1} + n^n, n \in N$$
I mean for $S = f(n)$, is there a formula to compute $f(n)$?
| I don't know if a closed form exists, but an asymptotic approximation can be given as
$$
\begin{align}
&n^n\left(1+\frac1n\frac{(n-1)^{n-1}}{n^{n-1}}+\frac1{n^2}\frac{(n-2)^{n-2}}{n^{n-2}}+\frac1{n^3}\frac{(n-3)^{n-3}}{n^{n-3}}+O\left(\frac1{n^4}\right)\right)\\
&=\bbox[5px,border:2px solid #C0A000]{n^n\left(1+\frac1{e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Let $a, b$ and $c$ be the lengths of the sides of an arbitrary triangle. Pick out the true statements. Let $a, b$ and $c$ be the lengths of the sides of an arbitrary triangle. Define
$$x =\frac{ab + bc + ca}{a^2 + b^2 + c^2}.$$
Pick out the true statements.
(a) $1/2 ≤ x ≤ 2$.
(b) $1/2 ≤ x ≤ 1$.
(c) $1/2 < x ≤ 1$. ... | We know in every triangle $ABC$ there are some useful relations called Law of cosines:
$$a^2=b^2+c^2-2bc\cos(A)\\b^2=a^2+c^2-2ac\cos(B)\\c^2=a^2+b^2-2ab\cos(C)$$ By adding them we have: $$a^2+b^2+c^2=2(bc\cos(A)+ac\cos(B)+ab\cos(C))$$ and if we take $A=B=C=60^{~\text{o}}$ then $x=1$($ABC$ is a Equilateral).
Now take $A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$ This question is from [Number Theory George E. Andrews 1-1 #3].
Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$
This problem is driving me crazy.
$$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$
$(x^n-y... | Let $u_n$ $=$ $x^n-y^n$. Now note that $u_n$ $=$ $(x+y)u_n$$_-$$_1$ $+$ $xy$ $u_n$$_-$$_2$. Assume that the given expression is true for all numbers from $1$ to some fixed $k(>1)$. Then apply induction to prove the result for $(k+1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/283056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Limit for a Recurrence Relation How I can find a limit for this recursively defined sequence?
$$a_0>0, a_{n+1}=\frac{a_{n}+2}{3a_{n}+2}$$
I'm particularly interested in answers involving concepts like contractive sequences and fixed points.
Many thanks.
| A little playing shows that the limit "should" be $\frac{2}{3}$. So it is natural to compute $a_{n+1}-\frac{2}{3}$.
We get
$$a_{n+1}-\frac{2}{3}=\frac{a_n+2}{3a_n+2}-\frac{2}{3}=\frac{\frac{2}{3}-a_n}{3a_n+2}.$$
Thus
$$\left|a_{n+1}-\frac{2}{3}\right|=\left|a_n-\frac{2}{3}\right|\frac{1}{3a_n+2}.$$
In particular,
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 2
} |
Inequality: $(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27$
Let be $a,b,c \geq 0$ such that: $a^2+b^2+c^2=3$.
Prove that:
$$(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27.$$
I try to apply $GM \leq AM$ for $x=a^3+a+1$, $y=b^3+b+1,z=c^3+c+1$ and
$$\displaystyle \sqrt[3]{xyz} \leq \frac{x+y+z}{3}$$ but still nothing.
Thanks :-)
| General comment: as soon as one tries to use AM-GM for
$x=a^3+a+1,$ $y=b^3+b+1$ and $z=c^3+c+1$ the inequality becomes wrong, since
$$a^3+a+b^3+b+c^3+c+3\ge 2(a^2+b^2+c^2)+3=9.$$
Using Lagrange multiplayers, one can reduce this problem to the following system:
$$(3a^2+1)(b^3+b+1)(c^3+c+1)=2\lambda a$$
$$(3b^2+1)(a^3+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
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Find the values of the positive constants $k$ and $c$ such that $-37\le k(3\sin\theta + 4\cos\theta) +c\le 43$ for all values of $\theta$ Hi how do i go about solving this?
Find the values of the positive constants $k$ and $c$ such that $$-37\le k(3\sin\theta + 4\cos\theta) +c\le 43 $$for all values of $\theta$ $$\righ... | By Cauchy-Schwarz
$$(3\sin(x)+4\cos(x))^2 \leq 25 (\sin^2(x)+\cos^2(x))=25$$
and equality is possible.
Then
$$-5 \leq 3\sin(x)+4\cos(x) \leq 5 $$
this shows that
$$-5k+c \leq - k(3\sin\theta + 4\cos\theta) +c\le 5k+c \,.$$
and the lower/upper bounds can be atatined. You can finish it easely.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the nth term of a recursive sequence I have a the following sequence:
$$\begin{gather}
a_1 = 3 \\
a_{n + 1} = 1 + \frac{a_n}{2}
\end{gather}
$$
How can I find the $a_n$ term?
| Put $n = 1$; you get $a_2 = 1 + \frac{1}{2} * 3 = \frac{5}{2}$
Following this continually, you get:
$$a_{n} = \frac{3 + \Sigma {2^{n - 1}}}{2^{n - 1}}$$
where, $n = 1, 2, 3 .. $
Aletnatively
You have $a_2 = 1 + \frac{a_1}{2}$
$$a_3 = 1 + \frac{a_2}{2} = 1 + \frac{1}{2} * (1 + \frac{a_1}{2}) = 1 + \frac{1}{2} + \frac{a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove $|\frac{1}{n} - \frac{1}{x}| \le \frac{1}{n^2}$ Prove $|\frac{1}{n} - \frac{1}{x}| \le \frac{1}{n^2}$ for all $n \le x \le n+1$, using the mean value theorem applied to $f(x) = \frac{1}{x}$
Immediately, I can recognize some components of the mean value theorem. $\frac{1}{n^2}$ likely comes from the slope of $f$ ... | $$f(x)=\frac{1}{x}\,\,\,\text{on the interval}\,\,\,[n\,,\,x]\,\,,\,n\leq x\leq n+1\Longrightarrow$$
$$\frac{f(x)-f(n)}{x-n}=f'(c)\,\,,\,\,\text{for some}\,\,\,c\in(n\,,\,x)\Longleftrightarrow$$
$$\frac{1}{x}-\frac{1}{n}=-\frac{1}{c^2}(x-n)$$
But
$$\left|\frac{1}{c^2}(x-n)\right|\leq\frac{1}{c^2}\Longleftrightarrow |x-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/288580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
log sin and log cos integral, maybe relate to fourier series I try to use the method of differentiation under integral sign for the first one
And integrate it back, but I failed to find the constant $c$ ....
Anyone hav other method?
$$\begin{align}
& \int_{0}^{\frac{\pi }{2}}{{{x}^{2}}{{\ln }^{2}}\left( 2\cos x \rig... | From
\begin{equation}
\int_{0}^{\frac{\pi }{2}}\left( 2\cos \theta \right) ^{x}\cos y\theta
d\theta =\frac{\pi }{2}F(1+\frac{x+y}{2},1+\frac{x-y}{2}), \tag*{(1)}
\end{equation}
where $F(x,y)=\frac{\Gamma (x+y-1)}{\Gamma (x)\Gamma (y)},$ we can get that
\begin{equation}
\begin{array}{c}
\int_{0}^{\frac{\pi }{2}}\theta ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/289587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Help solving summation series of a recursive function Yesterday in class, we were analyzing the Karatsuba multiplication algorithm and how it applies to recurrence equations. Time ran short, and I feel I missed how to solve the final summation.
First, we defined the recurrence equation as
$$T(n) = 3T \left(\frac{n}{2}... | The crucial mistake is that you passed somehow from
$$
(\frac32)^{(\log_2 n) +1}
$$
to
$$
3^{(\log_2 n )+ 1}.
$$
Using the rough formula that
$$
\sum_{x=0}^n r^x = O(r^n) \qquad \qquad\text{if $r>1$},
$$
you should get
$$
O(n (\frac32)^{\log_2 n})=O(n e^{(\ln n) \frac{\ln \frac32}{\ln 2}}) = O(n^{1+\frac{\ln \frac32}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/291681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Finding range of a function Can I get the range of the following function?
$$y=\dfrac{x^2-3x+2}{x^2-5x+6}$$
I cannot isolate $x$ to get $x=f(y)$.
Thanking you in advance.
| The function $ f(x) = \dfrac{x^{2} - 3x + 2}{x^{2} - 5x + 6} $ has domain $ (- \infty,2) \cup (2,3) \cup (3,\infty) $. By eliminating the common factor $ x - 2 $ in the numerator and denominator, we see that it coincides with the function $ g(x) = \dfrac{x - 1}{x - 3} $ on $ \text{Dom}(g) = (- \infty,3) \cup (3,\infty)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Elementary Set theory question .... it was asked in Exam There are $21$ people.
$9$ eat dish $A$
$10$ eat dish $B$
$7$ eat dish $C$
$5$ eat dish $A , B$ and $C$
How many people eat at least two dishes?
Answer:
$10$ (given in solutions)
$15$ (as per me )
Please tell me which one is correct. Also, tell if youu have dif... | There seems to be something wrong here.
Consider this Venn diagram and its labeled parts.
We are told that:
$$a+e+f+g=|A|=9 \qquad b+d+f+g=|B|=10 \qquad c+e+d+g=|C|= 7$$
$$g = 5$$
Thus
$$\begin{align}
a+e+f=4 \\
b+d+f=5 \\
c+e+d=2
\end{align}$$
so that, by adding all three equations,
$$a+b+c+2d+2e+2f=11 \qquad(1)$$
B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Find the sub determinant of a matrix If I have the matrix $A(x)$ which is $5 x 5$ and I need to find the $t x t$ sub-determinants of $A(x)$ for $t = 1$ to $t = 5$, how do I do this? $A(x)$ = \begin{pmatrix}
(x-1) & 0 & 0 & 0 & 0 \\
-1 & (x-1) & 0 & 0 & 0 \\
0 & 0 & (x-1) & 0 & 0\\
0 & 0 & 0 & (x+1) & 0\\
0 & 0 & 0 & ... | Let's look for the Smith Canonical Form, since it is your goal behind the question.
For $t=1$, we get in particular $x-1$ and $x+1$, whose $gcd$ is $1$, so the gcd of all $1\times 1$ minors is $1$.
For $t=2$, we get in particular $(x-1)^2$ and $(x+1)^2$, whose $gcd$ is $1$, so the gcd of all $2\times 2$ minors is $1$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trigonometry with multiple angle and exact value of $\tan\pi/5$ By considering the equation $\tan5\theta=0$, show that the exact value of $\tan\pi/5$ is $\sqrt{5-2\sqrt{5}}$.
Do I need to evaluate the multiple angle for $\tan5\theta=0$?
| The idea is to expand $\tan 5\theta$ so as to get a function of $\tan \theta$. Then by letting $\theta = \pi/5$, you will get an equation in $\tan \pi/5$.
In details, put $x = \tan \pi/5$ $$ \tan 5 \theta = \frac{\tan 2\theta + \tan 3\theta}{1 - \tan 2\theta \tan 3\theta}$$
Then you find that $\tan 2\pi/5 + \tan 3\pi/5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Induction on binomial Identity: $0\cdot {n\choose 0} + 2\cdot {n\choose 2} + 4\cdot {n\choose4}+\ldots = n\cdot2^{n-2}$ I am having trouble proving the following identity:
$0\cdot {n\choose 0} + 2\cdot {n\choose 2} + 4\cdot {n\choose4}+\ldots = n\cdot2^{n-2}$
Here is what I have so far:
Proof:
Base: Let $n=0$:
LHS: $0\... | Unlike with other proof methods, proofs by induction sometimes become easier if you make the statement to be proved stronger. In the present case, it seems that your approach should be successful if you add a corresponding statement about the sum of binomial coefficients with odd lower arguments to the claim:
$$
\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/295019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Prove using combinatorial argument $\binom{n}{0}.\binom{n}{2}+ \binom{n}{1}.\binom{n}{3}+\cdots+ \binom{n}{n-2}.\binom{n}{n}$ How can I prove it using combinatorial argument
$$
\binom n0\binom n2 + \binom n1\binom n3 + \cdots + \binom{n}{n-2}\binom nn = \binom {2n}{n-2}
$$
Sorry friends I have edited it
| Recall that $\binom{n}k=\binom{n}{n-k}$, so your identity can be written
$$\binom{n}0\binom{n}{n-2}+\binom{n}1\binom{n}{n-3}+\ldots+\binom{n}{n-2}\binom{n}0=\binom{2n}{n-2}\;.\tag{1}$$
Imagine that you have $n$ couples, each consisting of a man and wife, and you want to choose $n-2$ of these $2n$ people. Let $k$ be the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer Please help me with solving this :
prove that none of $\{11, 111, 1111 \ldots \}$ is the square of any $x\in\mathbb{Z}$ (that is, there is no $x\in\mathbb{Z}$ such that $x^2\in\{11, 111, 1111, \ldots\}$).
| Every integer is of one of the forms $\color{green}4k$, $\color{green}4k+\color{red}1$, $\color{green}4k+\color{red}2$ or $\color{green}4k+\color{red}3$ with $k$ integer. The square of an integer has therefore one of the forms
*
*$(\color{green}4k)^2 = 16k^2 = \color{green}4(4k^2) = \color{green}4k_0$,
*$(\color{gr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/298234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 10,
"answer_id": 7
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Induction Proof: $\sum_{i=1}^{n+1} i \cdot 2^i = n \cdot 2^{n+2}+2 $ Prove by Mathematical Induction . . .
$$\sum_{i=1}^{n+1} i \cdot 2^i = n \cdot 2^{n+2}+2 $$
for all $n \geq 0$
I tried solving it, but I got stuck near the end . . .
a. Basis Step:
$1\cdot 2^1 = 0\cdot 2^{0+2}+2$
$2 = 2$
b. Inductive Hypothesis
$$\... | You are done. All you need to do is to regroup the terms properly.
$$2k \cdot 2^{k+2} + 2^{k+3} + 2 = k \cdot 2^{k+3} + 2^{k+3} + 2 = (k+1) \cdot 2^{k+3} + 2$$
which is what you want.
As julien points out, your penultimate step must read $$k \cdot 2^{k+2} + 2 + \color{blue}{k \cdot 2^{k+2}} + 2^{k+3}$$
instead of
$$k ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integer solution to equation Does $$b =a-23\sqrt 3 \sqrt a+432$$ have infinitely many integer solutions?
$a = 3$ gives one;
$a = 27$ gives one;
$a = 27\times 9$ gives one.
| Let $a = 3n^2,\;$ where $n$ is any integer (and hence $\,a = 3n^2$ must also be an integer.)
So
$$
\begin{align}
b &= a-23\sqrt{3}\sqrt{a}+432 \\ \\
&= a -23\sqrt{3a}+ 432 \\ \\
&= (3n^2)-23\sqrt{3^2n^2}+432\\&=3n^2-23\cdot3n+432\\ \\
\iff b &= 3(n^2 - 23n + 144)
\end{align}
$$
Hence, for $a = 3n^2,\;b = 3(n^2 - 23n ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Another improper integral Show that :
$$\int_0^1\frac{(\sin ^{-1}x)^2}{x}\text{d}x=\frac{\pi ^2\ln 2}{4}-\frac78\zeta(3)$$
This integral is in "irresistible integrals" on page 122. I can't prove this one.
| This is not an answer, but you might be interested in knowing why this can work. I thought of writing this as a comment, but it's quite big.
*
*Note that $\displaystyle \ln(1+x) = x-\frac{x^2}{2} + \frac{x^3}{3} + \cdots$
*So $\ln(2) = \displaystyle 1-\frac{1}{2} + \frac{1}{3} + \cdots$
Now I tried the standard in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Finding all solutions to $ \tan^5x - 9\tan{x} = 0 $ I am stuck when it comes to finding the end value of a trig function. I have the following question:
$$ \tan^5x - 9\tan{x} = 0 $$
I worked the problem and got:
$$ \tan x = 0\\
\tan^4x-9 = 0\\
x = 0, \pi, \frac {\pi}{3}, \frac {2\pi}{3}, \frac {4\pi}{3}, \frac {5\pi}{3... | $\tan^2x=3\implies \cos2x=\frac{1-\tan^2x}{1+\tan^2x}=\frac{1-3}{1+3}=-\frac12=\cos\frac{2\pi}3$
$2x=2n\pi\pm \frac{2\pi}3=\frac{2\pi}3(3n\pm1)$ where $n$ is any integer.
$x=\frac{\pi}3(3n\pm1)$
Now, we need $0\le \frac{\pi}3(3n+1)<2\pi\implies 0\le3n+1<6\implies n=0,1$
For $n=0, x=\frac{\pi}3(3n+1)= \frac{\pi}3; n=1\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$ How to find the remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$
| As Ishan has suggested, $(2m+1)^2=4m^2+4m+1=8\frac{m(m+1)}2+1\equiv1\pmod8$
So, $\sum_{0\le r\le n}(2r+1)^2\equiv\sum_{0\le r\le n}1\pmod 8\equiv n+1\pmod8$
For $1013=2\cdot506+1,$ so $n=506\implies \sum_{0\le r\le 506}(2r+1)^2\equiv 507\pmod8\equiv3$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$a,b,c\in\mathbb{N}\:\land\:\frac ab+\frac bc+\frac ca\in\mathbb{N}\Rightarrow abc=n^3,\:n\in\mathbb{N}$
If $a$, $b$ and $c$ are natural numbers such that $$\frac ab+\frac bc+\frac ca\in\mathbb{N},$$ prove that $abc$ is a perfect cube.
All I could get was $ab^2+bc^2+ca^2\equiv_{abc}0$.
| Consider polynom $(X-\frac a b)(X- \frac b c)(X - \frac c a)$.
It is equal to $X^3 - u*X^2 + v*X - 1$, where $u = \frac a b + \frac b c + \frac c a$.
and $v = \frac b a + \frac c b + \frac a c$.
for $X = 1$, you get $1 - u + v - 1 = v - u$. So you have $(1-\frac a b)(1- \frac b c)(1- \frac c a) = v-u$.
You can rewrite ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
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How can I find the possible values that $\gcd(a+b,a^2+b^2)$ can take, if $\gcd(a,b)=1$ If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them.
| $$ \gcd(a+b,a^2+b^2) \mid \gcd((a+b)(a-b), a^2+b^2) = \gcd(a^2-b^2, a^2+b^2) \mid \gcd [ ( a^2+b^2)+ (a^2-b^2) , ( a^2+b^2)+ (a^2-b^2) ]=2 \gcd(a^2,b^2)=2$$
Now it is easy to check that both 1 and 2 are possible...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$ Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$.
I know the shortest distance exists between the curves on the common normal line. Is there any other shorter way to attempt?
| As the first curve is a circle you only have to find its center, and than compute the normal equations of your second curve. the one which goes through the center of the circle will do the job.
$$
\begin{align*}9x^2 + 9y^2 -30y+16 &= 9 \left( x^2+y^2 - \frac{10}{3}y +\frac{16}{9}\right)\\
&=9 \left(x^2+y^2-2\cdot \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all integers such that $\frac{n^3-3}{n^2-7}$ is an integer Find all integers such that $\frac{n^3-3}{n^2-7}$ is an integer.
I have no idea how to approach these types of proofs.
But I tried a few things, did not get me anywhere.
$n^3 -3 = an^2-7a$ then $n^3-an^2 = 3-7a$, and hence $n^2(n-a) = 3-7a$
And then I have... | Hint $\rm\ \ n^2\!-\color{#0A0}7\mid n^3\!-\color{#C00}3\ \Rightarrow\ mod\,\ n^2\!-7\!:\,\ \color{#0A0}7^3\! = n^6 = \color{#C00}3^2\:\Rightarrow\: n^2\!-7\mid 7^3\!-3^2 =\, 2\cdot 167\ $
$\rm\,2,\,167\,$ are prime so, by unique factorization, $\rm\:n^2\!-7\mid\, 2\cdot 167\,$ $\Rightarrow$ $\rm\, n^2\!-7 = \pm\{1, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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All sequences constructed by using the denominator as nominator and the sum of denominator and nominator as denominator converges to $\phi-1$ Assume we are given any number a. Write it in the form $a = \frac{b}{c}$ (if rational, in the usual way, if irrational, use forms like $\frac{a}{1}$).
Construct a sequence $\frac... | Given $\displaystyle\frac{F_{n+1}}{F_n}\to\varphi$ and $\displaystyle\frac{F_{n-1}}{F_n}\to\varphi^{-1}$, your sequence converges as
$$\frac{1/F_n}{1/F_n}\cdot\frac{F_{n-1}b+F_nc}{F_nb+F_{n+1}c}=\frac{b\displaystyle\frac{F_{n-1}}{F_n}+c}{\displaystyle b+c\frac{F_{n+1}}{F_n}}\longrightarrow \frac{b\varphi^{-1}+c}{b+c\va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Finding the largest possible value for the number of elements If $S$ is a subset from the set {$1,2,3,...,20$} such that the sum of any two elements is not divisible by $5$.How to find the largest possible value for the number of elements for $S$
| If we choose at least one number of the form $5a+1,$ we can not choose any number of the form $5b+4$ and vice versa where $a,b$ are non-negative integers.
If we choose at least one number of the form $5c+2$ we can not choose any number of the form $5d+3$ and vice versa where $c,d$ are non-negative integers.
Clearly,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given coordinates of center, equation of tangent to a circle, what is their point of contact? A circle centre $(-6,3)$ has a tangent with equation $$3X + 2Y = 40$$ What are the coordinates of the point of contact of the tangent with the circle?
| The radius of the circle must be perpendicular to the tangent. Rewriting $3x+2y=40$ gives us $y=-\frac{3}{2}x+20$, so the slope of the tangent line is $-\frac{3}{2}$. The slope of the perpendicular is the negative reciprocal, so the slope of the radius is $\frac{2}{3}$, and therefore the equation of the line the radius... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$ I need to show that
$$
\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}
$$
I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result which I am supposed to start from. Using a change of variable... | HINT: Use the relation
$$\int_0^\infty \left(\frac{\sin x}{x}\right)^n \mathrm{d}x
= \frac{\pi}{2^n (n-1)!}
\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k {n \choose k} (n-2k)^{n-1}$$
You may find a proof here A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
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"answer_id": 1
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Possible average square value suppose the sum of seven Positive number is 21. what is the minimum possible value of the average of the square of these number?
| Let's assume we have a 7-tuple in which some numbers are not equal. Call two numbers that are different $x$ and $y$, then $\frac {x^2+y^2} 2$ is more than $(\frac {x+y} 2 )^2$, which is the average if $x$ and $y$ were each replaced by their average, $(\frac {x+y} 2)$.To show this, first expand $(\frac {x+y} 2 )^2$ to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Limit of series involving ratio of two factorials $$
\sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3}
$$
The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. That's easy, but since we were given the limit, it got me thinking about h... | Notice that:
$$
\frac{j!^2}{(2j)!} = (2j+1) \operatorname{Beta}(j+1,j+1) = (2j+1) \int_0^1 t^j (1-t)^j \mathrm{d}t
$$
Now, interchanging the summation and the integration:
$$
\sum_{j=0}^\infty \frac{j!^2}{(2j)!} = \int_0^1 \sum_{j=0}^\infty (2j+1)(t(1-t))^j \mathrm{d} t = \int_0^1 \frac{1+t(1-t)}{(1-t(1-t))^2} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/319437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 0
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Solve the following homogeneous differential equation Initial value problem: $\displaystyle \frac{dy}{dx}= \frac{y}{x}+2 \frac{x^2}{y^2}$, $y(1)=1$. Can anyone help
| Again, start with $y = ux$ so that $\frac{dy}{dx} = u + x\frac{du}{dx}$ (applying the simple chain rule).
Substituting this into the original equation, along with
$ \frac{y}{x} = u $ and $ {\frac{x}{y}}^2 = u^{-2} $ results in
$u + x\frac{du}{dx} = u + 2u^{-2}$.
Since the u terms cancel on each side, the equation may ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$ where $|R_8(x)|\leq \frac{x^8}{8!}$ Prove that
$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$$
where $|R_8(x)|\leq \frac{x^8}{8!}$
| Hint: $f(x)=\sum_{i=0}^n\frac{f^ {(i)}(x)x^i}{i! }$
$\text if$ $$|f^{n+1}(x)|\le M_{n+1}$$then $$|R(x)|\le \frac{M_{n+1}|x|^n}{(n+1)!}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/323910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
finding the derivative using quotient rule and product rule find dy/dx;
a) $\frac{1-2x}{\sqrt{2+x}}$
b.) $3x(1-x^2)^{1/3}$
My attempt at a)
use the quotient rule:
so $dy/dx = -2 \sqrt{2+x}+ (1-2x)0.5(2+x)^{-1/2}$
but then I get stuck there, cannot simplify it, wolfram gives a nice simplified answer but not sure how to ... | a) $$\left(\frac{1-2x}{\sqrt{2+x}}\right)'=\frac{-2\sqrt{2+x}-(1-2x)\frac{1}{2\sqrt{2+x}}}{2+x}=\frac{-4(2+x)-(1-2x)}{2\sqrt{2+x}(2+x)}$$
$$=\frac{-9-2x}{2\sqrt{2+x}(2+x)}=-\frac{9+2x}{2(2+x)^{3/2}}$$
b.) $$(3x(1-x^2)^{1/3})'=3(1-x^2)^{1/3}-2x^2(1-x^2)^{-2/3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/326405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find the area of the Rose's petal.
If a Rose leaf is described by the equation $r = \sin 3\theta$, find the area of one petal.
| A sketch is useful here, but the only important observation is that $r=0$ when $\theta=0$, and again at $\frac{\pi}{3}$. These are your limits for one petal.
Since the area of a polar curve between the rays $\theta=a$ and $\theta=b$ is given by $\int_{a}^{b}\frac{1}{2}r^{2}d\theta$, we have
$$A=\int_{0}^{\pi/3}\frac{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there any procedure to compute the root of a matrix equation?? Actually I want to know the procedure, how could we calculate the value of $A$ where
$A^2$= $\begin{pmatrix} x & y\\ z & w\\ \end{pmatrix}$
| $A=\begin{pmatrix} x & y\\ z & w\\ \end{pmatrix}$
You want to find $B=\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}$ so that $B^2=A$
You can just calculate $B^2$ and solve for $a,b,c,d$ but a better way of doing it is to notice that $A$ commutes with $B$: $AB=B^2B=B^3=BB^2=BA$
From that, you can calculate $AB$ and $BA$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find the $f\left(x_i\right)$ for midpoint Riemann sum
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\mbox{Time} & \mbox{9am} & \mbox{10:30 am} & \mbox{Noon} & \mbox{1:30 pm} & \mbox{3 pm} & \mbox{4:30 pm} & \mbox{6 pm} & \mbox{7:30 pm} & \mbox{9 pm} \\
\hline
t & 0 & 1.5 & 3 & 4.5 & 6 & 7.5 & 9 & 10.5 & 12 \\
\h... | The midpoint of $[0,3]$ is $x_1=1.5$, so $f(x_1)=728$. The next interval to consider is $[3,6]$ with midpoint $x_2=4.5$. I think you can continue.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find all solutions of $x^8 \equiv 5 \pmod {3\cdot 11}$ Find all solutions of $x^8 \equiv 5 \pmod {3\cdot 11}$
Im not sure should I use primitive root or quadratic residue.
For primitive root, $U_{33} = \{1,2,4,5,7,8,10,13,14,16,17,19,20,23,25,26,28,29,31,32\}$
Thus $\phi_{(3\cdot11)}=20=2^2\times{5}$
But it takes way ... | $$x^8 \equiv 5 \pmod{33} \implies x^8 \equiv 5 \pmod 3 \implies (x^4)^2 \equiv 2 \pmod 3 \implies \text{No solution}$$
In general, when you want to solve for $$x^m \equiv a \pmod n \,\,\, (\spadesuit)$$ and if $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, the idea is to first solve for $$x^m \equiv a \pmod {p_l^{a_l}} \,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluate a big expression. If:$$\lambda=\int_{0}^{1}\frac{dx}{1+x^3} ;$$
Then evaluate : $$ p=\lim_{n \rightarrow \infty}\left( \frac{\prod_{r=1}^{n}\left(n^3+r^3\right)}{n^{3n}} \right)^{1/n} ;$$
EDIT: I have now ended upto $$lnp=\int_{0}^{1} \ln(1+t^3).dt$$
Now, we have to evaluate this using $\lambda$ , howsoever .
| First, to evaluate $\lambda$, use partial fractions:
$$\frac{1}{1+x^3} = \frac{1}{3} \left ( \frac{1}{1+x} - \frac{x-2}{x^2-x+1} \right )$$
The second piece in the parentheses is more amenable to integration when expressed as follows:
$$\frac{x-2}{x^2-x+1} = \frac{x-1/2}{(x-1/2)^2+3/4} - \frac{3/2}{(x-1/2)^2+3/4}$$
Now... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Showing series expansion How would I show $$\frac{\sinh z}{z^2}=\frac{1}{z}+\sum_0^\infty \frac{z^{2n+1}}{(2n+3)!}$$
I have $\sinh z=\sum_0^\infty\frac{z^{2n+1}}{(2n+1)!}$ and if I multiply by $\frac{1}{z^2}$ then $\frac{1}{z^2}\sinh z=\sum_0^\infty \frac{1}{z^2}\frac{z^{2n+1}}{(2n+1)!}$ but I how would I would factor... | So far you have
$$\frac{\sinh z}{z^2} = \sum_{n = 0}^\infty \frac{1}{z^2} \frac{z^{2n+1}}{(2n+1)!} = \sum_{n = 0}^\infty \frac{z^{2n-1}}{(2n+1)!} = \frac{1}{z} + \sum_{n = 1}^\infty \frac{z^{2n-1}}{(2n+1)!}.$$
If we introduce a new index $k = n - 1$, we have
$$\frac{\sinh z}{z^2} = \frac{1}{z} + \sum_{k = 0}^\infty \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove Vandermonde's Identity: $\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$? How can we prove that
$$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$
(Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Probability, 8th ed by Sheldon Ross.
| Consider the $K\times K$ matrix
$$B= \left[ \begin{array}{ccccccc}
1 & 0 & 0 & \cdots & 0 & 0 & 0 \\
1 & 1 & 0 & \cdots & 0 & 0 & 0 \\
0 & 1 & 1 & \cdots & 0 & 0 & 0 \\
0 & 0 & 1 & \cdots & 0 & 0 & 0 \\
& & & \ddots & & & \\
0 & 0 & 0 & \cdots & 1 & 1 & 0 \\
0 & 0 & 0 & \cdots & 0 & 1 & 1 \\
\end{array}
\right]$$
When... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Proof: How to prove $n$ is odd if $n^2 + 3$ is even New to the whole proof thing.
Trying to figure out that, for all integers $n$, if $n^2 + 3$ is even, then $n$ is odd.
Thank you for the help.
| The notation "$m\mid n$" means "$m$ divides $n$," or "$n$ is divisible by $m$." In contrast, $m\not\mid n$ means "$m$ does NOT divide $n$," or put differently, "$n$ is not divisible by $m.$"
Assume $n^2 + 3$ is even.
$(1)$ $n^2 + 3$ is even means, by definition, that $2$ divides $n^2 + 3$:
$$2\mid (n^2 + 3) \iff n^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/341165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
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Mersenne numbers congruent to two Prove that $2^{2^n-1} = 2 \pmod{2^n - 1}$ given that $2^n = 2 \pmod n$.
How would I go about proving that? I started by saying let $m = 2^n - 1$
Then, $2^n = 1 \pmod{m}$.
So I need to prove $2^m = 1 \pmod m$. I'm not sure how to proceed.
| Observe that $2^n \equiv 1 \pmod{ 2^n-1} $.
Given that $2^n \equiv 2 \pmod{n}$, we know that $2^n - 1 \equiv 1 \pmod {n}$, or that $2^n -1 = kn + 1 $ for some integer $k$.
Thus, $2^{ 2^n -1 } = 2^{kn + 1} \equiv 2 \pmod{2^n-1}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When computing the Legendre symbol $(\frac{3}{p})$, how to combine different congruences together into one statement? I try to find Legendre symbol for $\left(\dfrac{3}{p}\right)$.
This is what I did so far:
case 1: $p=3\mod{4}. $
so $\left(\dfrac{3}{p}\right)=-\left(\dfrac{p}{3}\right)$. now $\left(\dfrac{3}{p}\right... | As you've noted, the cases are as follows:
*
*If $p\equiv 3\bmod 4$, then $(\frac{3}{p})=-(\frac{p}{3})$.
*
*If $p\equiv 1\bmod 3$, then $(\frac{p}{3})=1$, so $(\frac{3}{p})=-1$.
*If $p\equiv 2\bmod 3$, then $(\frac{p}{3})=-1$, so $(\frac{3}{p})=1$.
*If $p\equiv 1\bmod 4$, then $(\frac{3}{p})=(\frac{p}{3})$.... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do we know the rank is 1? Here's matrix A:
$$
\begin{pmatrix}
2 & 1 \\
-4 & -2 \\
-2 & -1 \end{pmatrix}
$$
Apparently, we can determine that matrix A is of rank 1 by noting that.
$$
\det
\begin{pmatrix}
2 & 1 \\
-4 & -2 \end{pmatrix}
=
\det
\begin{pmatrix}
2 & 1 \\
-2 & -1 \end{pmatrix}
=
\det
\begin{pmatrix}
-4 & ... | The size of this matrix is $3\times 2$ so $\mathrm{rank}A\leq \min(2,3)=2$ and since the column $C_1=2C_2\not=0$ so
$$\mathrm{rank}A=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/345111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exponential and logarithmic series: Find the sum of $2^2 + 3^2/ 2!+4^2/3!+...$ to infinity
Find the sum of the following series:
$ 2^2 + 3^2/2! + 4^2/3! + ...$ to infinity
The answer is given as $5e$ but I got it as $5e+1$
$T_n = 1/(n-2)! +3/(n-1)! + 1/(n)! $ for $n \ge 2$
and $T_1+T_2 + ... $ to infinity = $4 + e... | You (after correction) : $\ \boxed{5\,e-1}$
\begin{align}
S(x):&=\sum_{k=1}^\infty \frac{x^{k+1}}{k!}&=x\,(e^x-1)\\
x\frac d{dx}S(x)&=\sum_{k=1}^\infty \frac{(k+1)\,x^{k+1}}{k!}&=x\,((x+1)e^x-1)\\
x\frac d{dx}\left(x\frac d{dx}S(x)\right)&=\sum_{k=1}^\infty \frac{(k+1)^2\,x^{k+1}}{k!}&=x\,((x^2+3x+1)e^x-1)\\\
\end{alig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Divide et impera recurrence, why induction does not work? $$
T(n) = T\left(\frac n2\right) + 2^n
$$
where $n \ge 1$ and $T(1) = 1$. If I understand the substitution method and the induction, I can guess that $T(n) = O(2^n)$.
I must prove that $T(n) = O(2^n)$, meaning constants $c$ and $n_0$ exist such that $T(n) \le c2... | From the definition,
$T(2) = T(1)+2^2 = 5$,
$T(3) = T(1)+2^3 = 9$,
and $T(4) = T(2)+2^4 = 21$.
So, for $1 \le n \le 4$, $T(n) < 2\cdot 2^n$.
Suppose $n > 4$ and $T(m) < c 2^m$ for
$n/2 \le m < n$.
Then, as before,
$T(n) = T(n/2)+2^n
< c 2^{n/2}+2^n
= 2^n(1+c 2^{-n/2})
< c 2^n
$
if
$1+c 2^{-n/2} < c$
or $1 < c(1-2^{-n/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/352568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Solve a linear equations with 3 unknowns I have this equations
$$x+3y+6z=3$$
$$x+y+z=-2$$
$$-x+y+4z=7$$
my solution is
$$x=0$$
$$y=-5$$
$$z=3$$
this task is 1 of 3 and there is going to be 1 that can be solved, 1 that will have a solution with rational numbers and the last 1 wont have any solution. This last one wont ... | As via Cramer's Rule,
$D=\det\begin{pmatrix} 1 & 3 & 6 \\ 1 & 1 & 1 \\ -1 & 1 & 4\end{pmatrix}=0,$
there will be a family of solutions
From the first two equations,
$$x+3y+6z-3=0$$
$$x+y+z+2=0$$
Solving for $x,y$ we get $x=\frac{3z-9}2,y=\frac{5-5z}2$
Putting the values of $x,y$ in the third equation, $$-\frac{3z-9}2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/352963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How can I solve this system? Having this system:
$$\begin{cases}
2 x(x^2 + y - 11) + x + y^2 - 7 = 0,\\
x^2 + y - 11 + 2 y (x + y^2 - 7) = 0 .
\end{cases}$$
How can I solve it without using a computer?.
Thank you!
| Let $a = x^2+y-11$ and $b = x+y^2-7$.
The equations we get are as follows:
$$2ax+b=0$$
$$a+2by=0$$
Replacing $a$ in the first equation yields:
$$-4bxy+b = 0$$
which gives us two options: $1-4xy = 0$ ($x = 1/4y$) or $b = 0$ ($x = 7-y^2)$. In both cases, we have expressed $x$ explicitly, we can insert it back the second ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\cos(A+B)$, given $\cos A$ and $\sin(B)$ Find the exact value:
Find $\cos(A+B)$ given that $\cos A=1/3$, with $A$ in the first quadrant, and $\sin B = -1/4$, with $B$ in the fourth quadrant.
| Remember that cosine represents adjacent over hypotenuse, and sine represents opposite over hypotenuse.
$\sin (A) = -\dfrac {1}{4} $ is therefore tells us that we have a triangle with a side length of one, and a hypotenuse of length four. Using Pythagorean theorem, we can find the third side, which is equal to $\sqrt {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
MacLaurin series of $\ln(1-x^2)$
The MacLaurin series for $\ln(1 + x)$ is obtained from the series for $\frac{1}{1 + x}$
by integration. Use this and appropriate substitutions to obtain the MacLaurin series for $\ln(1-x^2)$.
I did $$\int\frac{2}{x-\frac{1}{x}}$$ but I'm not sure how I'd go about the rest of the su... | I think what the question is asking is to use
$$
\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+\dots\tag{1}
$$
then substitute $x\mapsto-x$
$$
\log(1-x)=-x-\frac{x^2}2-\frac{x^3}3-\frac{x^4}4-\dots\tag{2}
$$
Then using the identity that $\log(ab)=\log(a)+\log(b)$ to get
$$
\begin{align}
\log(1-x^2)
&=-2\frac{x^2}{2}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
How to solve for $y$ in $x = \frac{\sqrt[3]{9y-5}}{\sqrt[3]{11}}$ I am trying to solve the following for $y$ but am lost. I tried to multiply by $\sqrt[3]{121}/\sqrt[3]{121}$ but don't think that is how to do it.
$$x = \frac{\sqrt[3]{9y-5}}{\sqrt[3]{11}}$$
| $$x = \frac{\sqrt[3]{9y-5}}{\sqrt[3]{11}}$$
Cube both sides.
$$x^3=\frac{9y-5}{11}$$
Multiply both sides by $11$.
$$11x^3=9y-5$$
Add $5$ to both sides.
$$11x^3+5=9y$$
Divide both sides by $9$.
$$y=\frac{11x^3+5}{9}$$
Very easy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/358896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
inverse trigonometry derivation Prove that :
$sin^{-1}x+sin^{-1}y = sin^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$
If -1 $\leq x \leq 1; -1 \leq y \leq 1 $ and $x^2+y^2\leq 1$ or if $xy <0 $ and $x^2+y^2 > 1$
solution :
Let $sin^{-1}x = \theta$ and $sin^{-1}y = \alpha$
Since sin($\theta + \alpha) = sin\theta cos\alpha + co... | As $x=\sin\theta, \cos^2\theta=1-x^2\implies 1-x^2\ge 0\implies -1\le x\le 1$
Similarly, $-1\le y\le 1$
As the principal value of $\arcsin x$ lies in $\in[-\frac\pi2,\frac\pi2]$
the given relationship will hold iff $-\frac\pi2\le \arcsin x+\arcsin y\le \frac\pi2$
$(1)$ If $x\cdot y<0$ let $x>0,y<0$
So, $0\le \arcsin ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$ Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$
I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
| Here is another way using Fourier analysis:
Let
\begin{equation*}
f(t)=\begin{cases}
t-t^2 & 0<t<1 \\
-f(-t) & -1 < t < 0
\end{cases}
\end{equation*}
be a function with period 2. Then we can express $f$ in a Fourer series:
\begin{equation*}
f(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n \cos n\pi t+b_n \sin n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 7,
"answer_id": 3
} |
Is my textbook wrong? My textbook says (without explaining how it is done):
$$\begin{pmatrix}
1\ 2\ 3\ 4\\
2\ 1\ 4\ 3
\end{pmatrix}\begin{pmatrix}
1\ 2\ 3\ 4\\
2\ 3\ 4\ 1
\end{pmatrix}=\begin{pmatrix}
1\ 2\ 3\ 4\\
3\ 2\ 1\ 4
\end{pmatrix},$$
and as I understand this is obtained by doing calculatio... | Permutations are functions, so most people do the evaluation from right to left.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/360305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
How prove this $(abc)^4+abc(a^3c^2+b^3a^2+c^3b^2)\le 4$ let $a,b,c>0$,and such that $a+b+c=3$,prove that
$$(abc)^4+abc(a^3c^2+b^3a^2+c^3b^2)\le 4$$
I first consider
$$abc\le\left(\dfrac{a+b+c}{3}\right)^3=1$$
so it suffices to show that
$$a^3c^2+b^3a^2+c^3b^2\le 3$$
But I find this not true.
| Let $\{a,b,c\}=\left\{x,y,z\right\}$, where $x\geq y\geq z$.
Hence, by Rearrangement and AM-GM we obtain:
$$a^2c+b^2a+c^2b+abc=a\cdot ac+b\cdot ba+c\cdot cb+xyz\leq x\cdot xy+y\cdot xz+z\cdot yz+xyz=$$
$$=y(x+z)^2=4x\left(\frac{x+z}{2}\right)^2\leq4\left(\frac{y+\frac{x+z}{2}+\frac{x+z}{2}}{3}\right)^3=4.$$
Thus, since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
convolution computation involving $e^{-x^2}$ In working a problem involving convolution, I have arrived at the following integral, but do not know how to compute it:
$$2\int_0^{\infty}e^{-a(x-y)^2-by^2}dy$$
I thought that this integrand did not have an antiderivative. Based on the way the problem is formulated, however... | Expand the argument of the exponential:
$$\begin{align}\int_0^{\infty} dy \: e^{-a (x-y)^2} e^{-b y^2} &= e^{-a x^2} \int_0^{\infty} dy \: e^{-(a+b)y^2 + 2 a x y}\end{align}$$
Now complete the square in the exponential:
$$\begin{align}(a+b) y^2 - 2 a x y &= (a+b) \left[y^2 - \frac{2 a x}{a+b} y + \left( \frac{a x}{a+b}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Expected Number of Coin Tosses to Get Five Consecutive Heads A fair coin is tossed repeatedly until 5 consecutive heads occurs.
What is the expected number of coin tosses?
| Here is a generating function approach.
Consider the following toss strings, probabilities, and terms
$$
\color{#00A000}{
\begin{array}{llc}
T&\frac12&\qquad\frac12x\\
HT&\frac14&\qquad\frac14x^2\\
HHT&\frac18&\qquad\frac18x^3\\
HHHT&\frac1{16}&\qquad\frac1{16}x^4\\
HHHHT&\frac1{32}&\qquad\frac1{32}x^5\\
\color{#C00000... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "102",
"answer_count": 13,
"answer_id": 3
} |
How to solve this determinant: $a_{ij}=|i-j|+1$? I have to solve determinant of the following form:
$$a_{ij}=|i-j|+1$$
It looks like this:
$$
\begin{pmatrix}
1 & 2 & 3 & 4 & \cdots & n \\
2 & 1 & 2 & 3 & \cdots & n-1 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
n & n-1 & n-2 & n-3 & \cdots & 1
\end{pmatri... | What do you get when you substract (i+1)th row - ith row?
$$
\begin{pmatrix}
1 & 2 & 3 & 4 & \cdots & n \\
1 & -1 & -1 & -1 & \cdots & -1 \\
1 & 1 & -1 & -1 & \cdots & -1 \\
1 & 1 & 1 & -1 & \cdots & -1 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 1 & 1 & 1 & \cdots & -1
\end{pmatrix}
$$
What do you ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.
Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$.
The contour is $\gamma=\gamma_1+\gamma_2+\gamma_3$ where $\gamma_1... | $$\frac{1}{x^3+1}=\frac{1}{3(x+1)}-\frac{x-2}{3(x^2-x+1)}$$
But
$$\frac{x-2}{x^2-x+1}=\frac{1}{2}\frac{2x-1}{x^2-x+1}-\frac{\frac{3}{2}}{\frac{3}{4}+\left(x-\frac{1}{2}\right)^2}=\frac{(x^2-x+1)'}{x^2-x+1}-\frac{4}{3}\frac{\frac{3}{2}}{1+\left(\frac{2}{\sqrt3}\left(x-\frac{1}{2}\right)\right)^2}=$$
$$=\frac{(x^2-x+1)'}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
An infinite fraction Here's a problem I saw on the AoPS twitter. I thought I might as well post it so that it could be discussed and a solution recorded.
What is the value of the following?
$$\cfrac{4}{{1 + \cfrac{1^2}{2 + \cfrac{3^2}{2 + \cfrac{5^2}{2 + \cfrac{7^2}{2 + \cfrac{9^2}{\ddots} }}}}}}$$
As an aside, I fin... | $$\arctan(x) = \cfrac{x}{1+\cfrac{1^2x^2}{3-x^2+\cfrac{3^2x^2}{5-3x^2+\cfrac{5^2x^2}{7-5x^2 + \cfrac{7^2x^2}{9-7x^2 + \ddots}}}}}$$
Take $x=1$ to get what you want.
EDIT
The proof for the above is the observation that if
$$S_n = a_1 + a_1 a_2 + a_1 a_2a_3 + a_1 a_2 a_3 a_4 + \cdots + (a_1 a_2 \cdots a_n)$$ then
$$S_n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Congruence modulo a prime. The goal is to show that $x^2+3\equiv 0 \pmod p$ is solvable for every prime $p$, $p\equiv 1 \pmod 3$.
What I know so far is that, since $3\mid (p-1)$, $x^3\equiv 1 \pmod p$ has exactly three solutions. From that you get that $x^3-1\equiv(x-1)(x^2+x+1) \pmod p$ and therefore $x^2+x+1\equiv... | This type of problem is commonly solved with Legendre symbols: $\left(\frac{-3}{p}\right)=\left(\frac{-1}{p}\right)\left(\frac{3}{p}\right)=(-1)^{\frac{p-1}{2}}(-1)^{\lfloor \frac{p+1}{6} \rfloor}$, which is $1$ if $p\equiv 1,7\pmod{12}$ and $-1$ if $p\equiv 5,11\pmod{12}$.
A more direct approach is to note that $-3$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Computing the Semimajor and Semiminor axis of an Ellipse I have the equation of the ellipse which is $\frac {x^2}{4r^2}+\frac{y^2}{r^2}=1$ Putting the (4,2) point on the ellipse we get $r^2=8$ so we get the equation $\frac {x^2}{32}+\frac {y^2}8=1$ and the semi-major axis is $\sqrt {32}= 4\sqrt 2$, the semi-minor axis... | The ellipse is in standard position and orientation, with the axes along the $x$ and $y$ axes. The ellipse meets the $x$-axis at the two points $x=\pm\sqrt{32}$, $y=0$, and meets the $y$-axis at $x=0$, $y=\pm\sqrt{8}$.
The distance from $(-\sqrt{32},0)$ to $(\sqrt{32},0)$ is bigger than the distance between the two $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Need help solving $\int x \sqrt{\frac {a^2 - x^2} { a^2 + x^2 }} dx$ I have a complicated integral to solve. Can someone provide a better way to solve it than what i did - dividing by a inside the root, and then putting $ t = x / a $, and then putting $t^2 = \cos \theta $ and so many other substitutions.
$$
\int x \s... | We can use the algebraic substitution:
$$t=\frac{a^{2}-x^{2}}{a^{2}+x^{2}}.\tag{0} $$
We have that
$$\begin{eqnarray*}
I &=&\int x\sqrt{\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}dx=-\int \sqrt{t}\frac{a^{2}
}{\left( 1+t\right) ^{2}}dt\tag{1} \\
&=&a^{2}\frac{\sqrt{t}}{1+t}-a^{2}\arctan \sqrt{t}+C \\
&=&\frac{a^{2}+x^{2}}{2}\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/371282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the values of $p$ such that $\left( \frac{7}{p} \right )= 1$ (Legendre Symbol)
Show that if $p$ is an odd prime coprime to $7$, then $\left( \frac{7}{p} \right) = 1$ if and only if $p \equiv \pm 1, \pm 3,$ or $\pm 9 \pmod{28}$. HINT: If $p$ is an odd prime, determine which values can $p$ take $\mod28$, and consid... | Here is a solution that avoids the casework suggested by Zev Chonoles. Note that the computation of $\left(\tfrac{-7}{p}\right)$ is much easier. Once we know this, the computation of $\left(\tfrac{7}{p}\right)$ falls to CRT. By the Quadratic Reciprocity law, $$\left(\frac{-7}{p}\right)=\left(\frac{-1}{p}\right)\left(\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/371942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Finding generating function for the recurrence $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$ I am trying to find generating function for the recurrence:
*
*$a_0 = 1$,
*$a_n = {n \choose 2} + 3a_{n - 1}$ for every $n \ge 1$.
It looks like this:
*
*$a_0 = 1$
*$a_1 = {1 \choose 2} + 3$
*$a_2 = {2 \choose 2} + 3... | A related problem. Assume $ F(x) = \sum_{n=0}^{\infty}a_n x^n $, then
$$ a_n = {n \choose 2} + 3a_{n - 1} \implies a_{n+1} = {n+1 \choose 2} + 3a_{n}$$
$$ \sum_{n=0}^{\infty} a_{n+1} x^n = \frac{1}{2}\sum_{n=0}^{\infty}n(n+1)x^n + 3\sum_{n=0}^{\infty}a_{n}x^n $$
$$ \implies \sum_{n=1}^{\infty} a_{n} x^{n-1} = \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/372439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Geometry: Hyperbolas I was wondering how would I complete the square for this particular hyperbola?
$4x^2 - 5y^2 + 24y = 16$
I tried this technique but to no avail:
$$4x^2 - 5(y^2 + \frac{24}{5}y) = 16$$
$$ \Rightarrow 4x^2 - 5(y + \frac{12}{5})^2 = 16 + \left(\frac{12}{5}\right)^2$$
$$\Rightarrow 4x^2 - 5(y + \frac{12... | You have wrongly added $\left(\frac{12}5\right)^2$ in place of subtraction
$(2x)^2-5\{y^2-2\cdot y\cdot \frac{12}5+ (\frac{12}5)^2\}=16- \left(\frac{12}5\right)^2$
or, $(2x)^2-5\left(y- \frac{12}5\right)^2=\frac{256}{25}=\left(\frac{16}5\right)^2$
and so on
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/372555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Recursive formula for variance I'd like to know how I can recursively (iteratively) compute variance, so that I may calculate the standard deviation of a very large dataset in javascript. The input is a sorted array of positive integers.
| Here are the iterative formulas (with derivations) for the population ($N$ normalized) and sample ($N-1$ normalized) standard deviations, which express the $\sigma_{n+1}$ ($s_{n+1}$ for sample) for the $n+1$ value set in terms of $\sigma_{n}$ ($s_{n}$ for sample), $\bar x_{n}$ of the $n$ value set plus the new value $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 2
} |
Solving an integral equation using the Fourier transform I have to solve the equation
$\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$
Using fourier transform. I know this is half of the usual fourier cosine transform, and so that I would get back $f(x)$ using $\frac{2}{\pi} \int_0^{\infty} \f... | $$ \begin{align} \frac{2}{\pi} \int_{0}^{\infty} \frac{\sin (\alpha) \cos (x \alpha)}{\alpha} \ d \alpha &= \frac{1}{\pi} \int_{0}^{\infty} \frac{\sin \big((1+x) \alpha \big)+\sin \big( (1-x) \alpha \big)}{\alpha} \ d \alpha \\ &= \frac{1}{\pi} \Big(\text{sgn}(1+x) \frac{\pi}{2}+\text{sgn}(1-x) \frac{\pi}{2} \Big) \\ &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/375960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
is $f$ analytic inside $C?$
If $$f(z_0)=\dfrac{1}{2\pi i}\int_C\dfrac{f(z)}{z-z_0}dz$$ for all point $z_0$ inside $C,$ is $f$ analytic inside $C?~(C:$ simple closed contour$)$
| For simplicity, assume $C:|z-z_0|=r$ be a circle around $z_0$ and $|f(z)|\leqslant M$ for all $z\in C$. Then
\begin{align*}
\Big|\frac{f(z_1)-f(z_0)}{z_1-z_0}\Big|
=&\Big|\frac{1}{z_1-z_0}\Big|\cdot\Big|\dfrac{1}{2\pi i}\int_C\dfrac{f(z)}{z-z_1}dz-\dfrac{1}{2\pi i}\int_C\dfrac{f(z)}{z-z_0}dz\Big|\\
=&\frac{1}{2\pi}\Bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Factorization of the trinomial $x^{2n}+Dx^n+1$? The following trinomials will factor for any $a$,
$$1+a(-3+a^2)x^3+x^6 = (1+ax+x^2)(1-ax-x^2+a^2x^2-ax^3+x^4)\tag{1}$$
and similarly for,
$$1+a(5-5a^2+a^4)x^5+x^{10}\tag{2}$$
$$1+a(-7+14a^2-7a^4+a^6)x^7+x^{14}\tag{3}$$
$$1+a(9-30a^2+27a^4-9a^6+a^8)x^9+x^{18}\tag{4}$$
and ... | If we work in ${\mathbf Z}[x,a]/(x^2+ax+1)$, with $a$ and $x$ as independent indeterminates then we seek a polynomial $D_n$ in ${\mathbf Z}[a]$ such that $x^{2n}+D_nx^n+1 \equiv 0 \bmod x^2+ax+1$ for odd $n$.
Starting from $x^2 \equiv -ax-1 \bmod x^2+ax+1$ we can compute higher powers of $x$ to that modulus by using th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/382342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $a+b+c=1$, find the minimum of $\frac{4+3abc}{ab+bc+ac}$ I came to ask this because I am really stuck at this problem. I have tried everything from arithmetic mean, geometric mean and harmonic mean. Also, I have tried playing with the variables and such, but it got me to nowhere.
If $a+b+c=1$; $a,b,c$ nonnegative, c... | Substitute $a=1-(b+c)$ in $\displaystyle\frac{4+3abc}{ab+bc+ac}$ to get
$$\frac{4+3(1-(b+c))bc}{(1-(b+c))b+bc+(1-(b+c))c} \tag{1}$$
Differentiating $(1)$ with respect to $b$ gives
$$\frac{\partial}{\partial b}\left(\frac{4+3(1-(b+c))bc}{(1-(b+c))b+bc+(1-(b+c))c}\right) =\frac{(3c^3-3c^2+4)(2b-c+1)}{\left(b^2+b(c-1)+c(c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/383918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
calculate the roots of $z = 1 + z^{1/2}$ using Lagrange expansion I am trying to find the roots of equation $z = 1 + \sqrt{z}$ using Lagrange's expansion given on this book page no 15.
I expanded $z = 1 + \zeta z^p$ as
$$z = 1 + \zeta + \frac{2p}{2!}\zeta^2 + \frac{3p(3p-1)}{3!}\zeta^3+\frac{4p(4p-1)(4p-2)}{4!}\zeta^... | You can solve the equation explicitly by squaring both sides then solving the resulting quadratic. Compare the root you find with sum of the first terms of the series you gave.
In more detail.
We have
$$
z - 1 = \sqrt{z} \quad \Longrightarrow \quad z^2 - 2z + 1 = z,
\tag{1}
$$
so that, by the quadratic formula, the tw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Showing that $2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\left( \frac{x^{a-1}}{\sinh x} - x^{a-2} \right) \mathrm dx$ I want to show that $$2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\Big( \frac{x^{a-1}}{\sinh x} - x^{a-2}\Big) \, \mathrm dx \, , \quad 0 < \Re ... |
$$ \color{blue}{\Gamma(s-N)\zeta(s-N)=\int_{0}^{\infty}x^{s-N-2}\left[\frac{x}{e^x-1}-\left(\sum_{n=0}^{N}B_{n}\frac{x^n}{n!}\right)\right]\,dx} $$
$$ {\small \,0\lt\,Re\{s\}\,\lt1 ,\quad N\in\{\,0,\,1,\,2,\,\cdots\,\} ,\quad B_{n}\,\,{Bernoulli\,Number} ,\quad B_{1}=-1/2} $$
$$
\begin{align}
I\,& =\,\int_{0}^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/385862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 2,
"answer_id": 1
} |
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Fractions in Questions and Answers
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