All Questions
249
questions
1
vote
0
answers
74
views
Finding the value of $ \sum_{n=1}^{\infty} \frac{2(2n+1)}{\exp( \frac{\pi(2n+1)}{2})+\exp ( \frac{3\pi(2n+1)}{2})}$
I have a question which askes to find the value of:
$$\displaystyle \tag*{} \sum \limits_{n=1}^{\infty} \dfrac{2(2n+1)}{\exp\left( \dfrac{\pi(2n+1)}{2}\right)+\exp \left( \dfrac{3\pi(2n+1)}{2}\right)}...
- 921
2
votes
4
answers
568
views
prove that $\ln(2) = \sum_{k=1}^\infty \dfrac{(-1)^{k+1}}{k}$ without using Abel's theorem [duplicate]
I know that
$$\ln(1+x) = \sum_{k=1}^\infty \dfrac{(-1)^{k+1}}{k}x^k$$
for $|x| < 1$. But how can I show
$$\ln(2) = \sum_{k=1}^\infty \dfrac{(-1)^{k+1}}{k}$$
without using Abel's theorem? There's a ...
- 1,225
3
votes
1
answer
66
views
How can i simplify the following formula: $\sum\limits_{i,j=1}^{n}(t_{j}\land t_{i})$?
Consider the following time discretization $t_{0}=0< t_{1} < ... < t_{n} = T$ of $[0,T]$ where the time increments are equal in magnitude, i.e. $t_{j}-t_{j-1}=\delta$.
How can i simplify the ...
- 1,838
3
votes
1
answer
55
views
Calculating a sum $\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}$
I want to calculate this sum, while $0<p<1$:
$$\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}$$
Is this correct:
$$\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}=\frac{1}{1-p}\cdot \frac{1-\frac{1}{(1-p)^k}}{1-\frac{1}{1-...
- 627
1
vote
1
answer
44
views
Calculating a sum sigma, with minimum $\sum_{i=1}^{\min(n,k-1)}\frac{1}{(1-p)^i}$
How do I calculate this sum:
$$\sum_{i=1}^{\min(n,k-1)}\frac{1}{(1-p)^i}$$
It is like a geometric sum, but it has the minimum which I do not know how to deal with.
I got this sum while calculating two ...
- 627
0
votes
0
answers
71
views
Find $\lim\limits_{n \to \infty}\frac{\ln^3 n}{\sqrt{n}}\sum_{k=2}^{n-2}\frac{1}{\ln k\ln(n-k)\ln(n+k)\sqrt{n+k}}$
On a Chinese website, someone has posted a solution as follows:
Notice
\begin{align*} s_n&:=\frac{\ln^3 n}{\sqrt{n}}\sum_{k=2}^{n-2}\frac{1}{\ln k\ln(n-k)\ln(n+k)\sqrt{n+k}}\\ &=\frac{1}{n}\...
- 13.9k
0
votes
1
answer
46
views
Problem amount!
There was a problem with the calculation of amounts
I converted the Main Amount:
\begin{align}
\sum\limits_{n=1}^{\infty }\frac{H^3_n}{2^n}&=\sum\limits_{n=1}^{\infty }\frac{1}{2^n}\left ( H_{n+1}-...
- 1,431
0
votes
0
answers
95
views
Proof of $sin$ formula.
I am reading this quesiton and accepted answer.
Question is about proof.
$S = \sin{(a)} + \sin{(a+d)} + \cdots + \sin{(a+nd)}$
$S \times \sin\Bigl(\frac{d}{2}\Bigr) = \sin{(a)}\sin\Bigl(\frac{d}{2}\...
- 879
4
votes
1
answer
129
views
Showing $\sum_{n=1}^\infty\frac1{n^2}\left(1+\frac1{2^2}+\cdots+\frac1{n^2}\right)=\frac{7\pi^4}{360}$
Help me calculate the amount in a simpler way:
\begin{aligned}
\mathcal{S}&=\sum\limits_{n=1}^{\infty }\frac{1}{n^2}\left ( 1+\frac{1}{2^2}+\ldots+\frac{1}{n^2} \right )=\sum\limits_{n=1}^{\infty ...
- 1,431
1
vote
0
answers
64
views
closed form for a particular sum of powers of $2$ or approximation
I was wondering if it's possible to find a closed form for the sum $S(n) := \sum_{i=1}^n 2^{2^{i-1}-1}$. I tried using induction but I'm not sure how to find a general formula for the pattern. Clearly ...
- 1,166
9
votes
4
answers
456
views
Limit of the series $\sum_{k=1}^\infty \frac{n}{n^2+k^2}.$
I am trying to evaluate $$\lim_{n\to \infty} \sum_{k=1}^\infty \frac{n}{n^2+k^2}.$$ Now I am aware that clearly $$\lim_{n\to \infty} \sum_{k=1}^n \frac{n}{n^2+k^2} = \int_0^1 \frac{1}{1+x^2}dx = \tan^{...
- 1,018
1
vote
2
answers
201
views
Determine if $\int_1^\infty \arctan(e^{-x})dx$ converges or diverges
I want to check if the sequence $\sum_{n=1}^\infty [\sin(n) \cdot \arctan(e^{-n})]$ converges absolutely, by condition, or not.
We know that $\sum_{n=1}^\infty [\sin(n) \cdot \arctan(e^{-n})] \le \...
- 627
1
vote
1
answer
94
views
Prove that equality between sums over one common factor and a distinct one entails equality of the distinct factors
This problem comes from a subfield of economics called decision theory, where we evaluate a decision from the distribution of outcomes it may entail according to a group of $N$ experts. According to ...
- 63
2
votes
3
answers
103
views
$\sum_{n=1}^{\infty} u_n^2=0$ $\Rightarrow$ $u_n=0 \ \forall \ n\in \mathbb{N}$
Let, $<u_n>$ be a real sequence and given that $$\sum_{n=1}^{\infty} u_n^2=0$$
Prove that $u_n=0 \ \forall \ n\in \mathbb{N}$
Attempt
$u_n^2\geq 0 \ \forall \ n\geq 1$
Since, $$\sum_{n=1}^{\...
user908035
1
vote
1
answer
64
views
Limit of product of kth roots
For $n\geq 1$ , let $$A(n)=\prod_{k=1}^{n}(1+\frac{k}{n})^\frac{1}{k}.$$ I'd like to determine $$ \lim_{n->\infty} A(n).$$ Taking logarithms and using the following crude bound gives $$\log(A(n))=\...
- 437