All Questions
1,809
questions
357
votes
8
answers
57k
views
Calculating the length of the paper on a toilet paper roll
Fun with Math time.
My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
- 26.3k
58
votes
7
answers
25k
views
Is it possible to write a sum as an integral to solve it?
I was wondering, for example,
Can:
$$ \sum_{n=1}^{\infty} \frac{1}{(3n-1)(3n+2)}$$
Be written as an Integral? To solve it. I am NOT talking about a method for using tricks with integrals.
But ...
- 11.2k
44
votes
1
answer
10k
views
Finding the sum- $x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$
If $S = x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$
Find S.
Note:This is not a GP series.The powers are in GP.
My Attempts so far:
1)If $S(x)=x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$
Then $$S(x)-S(x^{2})=x$$
...
- 5,078
42
votes
6
answers
7k
views
If $f(x)=\frac{1}{x^2+x+1}$, how to find $f^{(36)} (0)$?
If $f(x)=\frac{1}{x^2+x+1}$, find $f^{(36)} (0)$.
So far I have tried letting $a=x^2+x+1$ and then finding the first several derivatives to see if some terms would disappear because the third ...
- 621
42
votes
3
answers
1k
views
Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
- 13.2k
40
votes
2
answers
2k
views
Proving $\lim_{x \to 0+} \sum_{n=0}^\infty \frac{(-1)^n}{n!^x} = \frac{1}{2}$
Prove that $$ \lim_{x \to 0+} \sum_{n=0}^\infty \frac{(-1)^n}{n!^x} =
\frac{1}{2}. $$
We know that $$ \sum_{n=0}^\infty \frac{(-1)^n}{n!^x}$$ converges for any $x>0$. So I try to evaluate the ...
- 1,351
39
votes
12
answers
90k
views
Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$? [duplicate]
Can you please explain why
$$
\sum_{k=1}^{\infty} \dfrac{k}{2^k} =
\dfrac{1}{2} +\dfrac{ 2}{4} + \dfrac{3}{8}+ \dfrac{4}{16} +\dfrac{5}{32} + \dots =
2
$$
I know $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{...
- 433
38
votes
4
answers
3k
views
A closed form of $\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$
I am looking for a closed form of the following series
\begin{equation}
\mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)
\end{equation}
I have no idea how to ...
- 19.4k
37
votes
3
answers
3k
views
An inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$
$n$ is a positive integer, then
$$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$
please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$.
I want to find a ...
- 7,819
34
votes
3
answers
225k
views
First and second derivative of a summation
Consider the function $f(\mu) = \sum_{i = 1}^{n} (x_i - \mu)^2$, where $x_i = i,\,i=1, 2,\dots, n$.
What is the first and second derivative of $f(\mu)$?
- 4,181
34
votes
5
answers
2k
views
How find this sum $\sum\limits_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}$
Find this sum
$$I=\sum_{n=0}^{\infty}\dfrac{1}{(3n+1)(3n+2)(3n+3)}$$
My try: let
$$f(x)=\sum_{n=0}^{\infty}\dfrac{x^{3n+3}}{(3n+1)(3n+2)(3n+3)},|x|\le 1$$
then we have
$$f^{(3)}(x)=\sum_{n=0}^{...
- 93.6k
33
votes
1
answer
23k
views
Reversing the Order of Integration and Summation
I am trying to understand when we can interchange the order of Integration and Summation. I am increasingly encountering Integrals; some of which are being solved by interchanging the order of ...
- 4,116
32
votes
7
answers
14k
views
Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $
I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:
DEFINITION Given $x\in \Bbb R$, the integer ...
- 123k
32
votes
1
answer
818
views
On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$
Motivation
$\newcommand{Li}{\operatorname{Li}}$
It is already known that:
$$\Re\Li_2(1+i)=\frac{\pi^2}{16}$$
$$\Re\Li_3(1+i)=\frac{\pi^2\ln2}{32}+\frac{35}{64}\zeta(3)$$
And by this question, ...
- 8,679
29
votes
2
answers
829
views
How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$
How can we prove the following
$$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$
I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} \...
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