All Questions
46
questions
2
votes
1
answer
97
views
Find the interval of convergence $\sum_{n\geq1}\left(\ln\frac{1}{2}+1-\frac{1}{2}+\frac{1}{3}-\cdots-\frac{(-1)^n}{n}\right)x^n$
Find the interval of convergence A,of the $$\sum_{n\geq1}\left(\ln\frac{1}{2}+1-\frac{1}{2}+\frac{1}{3}-\cdots-\frac{(-1)^n}{n}\right)x^n$$
and calculate the sum for each value of $x\in A$ .
My work
$$...
user1254447
1
vote
1
answer
101
views
Compute ${\sum_{n=1}^{+\infty}(-1)^n\left(\sum_{k=1}^{n}\frac{1}{2k-1}-\frac{ \ln n}{2}-\frac{\gamma}{2}-\ln 2\right)}$
Compute
$${\sum_{n=1}^{+\infty}(-1)^n\left(\sum_{k=1}^{n}\frac{1}{2k-1}-\frac{ \ln n}{2}-\frac{\gamma}{2}-\ln 2\right)}.$$
What I have done so far
Lemma: $$\displaystyle{\mathop {\lim }\limits_{N \to \...
0
votes
1
answer
74
views
Closed form solution for partial summation of $\sum_{x=1}^{k} \frac{2^{\frac{1}{x}}}{{x^2}}$
Recently I've been working on solving summations and I found this one to be quite tricky.
$\sum_{x=1}^{k} \frac{2^{\frac{1}{x}}}{{x^2}}$
The integral which this is based off of, can be solved with u ...
3
votes
1
answer
86
views
Closed form of a geometric series without some term
I have to study find the closing form of the following series:
\begin{equation}
f(1) + f(2) + f(3) + f(5) + f(6) + f(7) + \dots
\end{equation}
So basically the sum of all terms without the multiple of ...
8
votes
6
answers
325
views
Calculate the closed form of the following series
$$\sum_{m=r}^{\infty}\binom{m-1}{r-1}\frac{1}{4^m}$$
The answer given is $$\frac{1}{3^r}$$ I tried expanding the expression so it becomes $$\sum_{m=r}^{\infty}\frac{(m-1)!}{(r-1)!(m-r)!}\frac{1}{4^m}$$...
user879314
15
votes
2
answers
865
views
Evaluate $\sum\limits_{n=1}^{\infty}\frac{1}{n^3}\binom{2n}{n}^{-1}$. [duplicate]
Evaluate $$\sum\limits_{n=1}^{\infty}\frac{1}{n^3}\binom{2n}{n}^{-1}.$$
My work so far and background to the problem.
This question was inspired by the second page of this paper. The author of the ...
- 6,560
0
votes
1
answer
326
views
(Calculus) Solving a geometric series word problem
I’m struggling with understanding how to solve part B of the following problem:
Consider an outdoor pool initially filled with 20,000 gallons of water. Each day, 4% of the water in the pool ...
- 11
0
votes
1
answer
72
views
some identity of summation and generalization
I see this way and idea from Simply Beautiful Art
profile here
Let : $S=\displaystyle\sum_{1≤k≤m≤n≤\infty}f(m)f(k)f(n)$
then :
$6S=\displaystyle\sum_{n,m,k≥1}f(m)f(k)f(n)+3\displaystyle\sum_{n,m≥...
- 2,323
7
votes
6
answers
242
views
Find triple summation rel in a closed form $S=\sum_{n=1}^{\infty}\sum_{m=1}^{n}\sum_{k=1}^{m}\frac{1}{(n+1)(k+1)(m+1)nmk}$
Evaluate $\displaystyle S=\sum_{n=1}^{\infty}\sum_{m=1}^{n}\sum_{k=1}^{m}\frac{1}{(n+1)(k+1)(m+1)nmk}$
My attempt :
Let $$A=\sum_{k=1}^{m}\frac{1}{k(k+1)}
=\sum_{k=1}^{m}\left( \frac1{k}-\frac1{...
- 2,323
6
votes
3
answers
169
views
evaluate the summation : $\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)(n+2x+3)}$
Find
$$S=\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)(n+2x+3)}$$
for $x≥0$.
At first, I use a partial fraction
$$S=\displaystyle\sum_{n=0}^{\infty}\left(\frac{(-1)^{n}}{2(x+1)(n+1)}-\frac{(-...
- 2,323
0
votes
2
answers
102
views
A Closed form for the $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3}$
I'm looking for a closed form for this sequence,
$$\sum_{n=1}^{\infty}\left(\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3} \right)$$
I applied convergence test. The series converges.I want to know ...
user548054
3
votes
1
answer
98
views
How can I sum the series $e^{-2}\frac{(3)^n}{n!}\sum_{k=0}^{\infty}\left ( \frac{1}{2}\right )^k\frac{1}{(k-n)!}$
How can I sum the following series?
$$e^{-2}\frac{(3)^n}{n!}\sum_{k=0}^{\infty}\left ( \frac{1}{2}\right )^k\frac{1}{(k-n)!}$$
I think I can make this sum in the form of exponential expansion but ...
- 1,204
5
votes
2
answers
200
views
What is the close form of: $\sum\limits_{k=1}^{\infty}\log\left(\frac{1}{k^2}+1\right)$
Is there a close form for of this series
$$\sum_{k=1}^{\infty}\log\left(\frac{1}{k^2}+1\right) =\log \prod_{k=1}^{\infty}\left(\frac{1}{k^2}+1\right)$$
I know it converges in fact since $ \log(x+1)\...
- 24.2k
1
vote
0
answers
116
views
Sum to closed form?
I need to evaluate the following summation to closed form:
$$\sum _{k=1}^{\infty } \frac{\sin (e k n \pi ) \sin (e k (1+n) \pi )}{k \pi \sin (e k \pi )}$$
where: $n>1$ and $n\in \mathbb{Z}$, $e \...
- 4,492
1
vote
5
answers
5k
views
Find a formula for $\sum_{i=1}^n (2i-1)^2 = 1^2+3^2+....+(2n-1)^2$
Consider the sum
$$\sum_{i=1}^n (2i-1)^2 = 1^2+3^2+...+(2n-1)^2.$$
I want to find a closed formula for this sum, however I'm not sure how to do this. I don't mind if you don't give me the answer but ...
- 11