All Questions
6
questions
3
votes
1
answer
135
views
An infinite sum of products
I have to calculate this sum in closed form
$$ \sum_{n=1}^\infty \prod_{k=1}^n \frac{x^{k-1}}{1 - x^k} $$
where $x < 1$.
Numerical evaluation shows that this converges. The product can be performed ...
- 86
1
vote
0
answers
56
views
If the Infinite sum of a series is known, what is the sum of element wise product with another series?
Suppose we know the summation of some series $G(n)$ such that, $$\sum_{n=1}^{\infty}G(n)=S_1.$$
Now assume another summation $S_2$ is expressed as,
$$S_2=\sum_{n=1}^\infty G(n) e^{i\frac{2\pi}{m}n}; \...
4
votes
2
answers
401
views
Infinite sum involving powers and factorials
I am interested in evaluating the following infinite sum
\begin{equation}
\sum_{n=0}^{\infty} \frac{\alpha^{n}}{n!}n^{\beta}
\end{equation}
where both $\alpha$ and $\beta$ are real numbers. However, ...
- 561
0
votes
3
answers
300
views
$\sum\limits_{n=4}^{\infty } \frac{2^n + 8^n}{10^n} = ?$
im looking for hints on how to do: $\sum\limits_{n=4}^{n= \infty }
\frac{2^n + 8^n}{10^n} = ?$
I thought this may have had something to do with geometric series but nothing obvious comes up
...
- 3,221
3
votes
2
answers
394
views
A "generalized" exponential power series
I'm wondering if
$$ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} $$
what would this be
$$ \sum_{k=0}^\infty \frac{x^{k+\alpha}}{\Gamma(k+\alpha)} = \large{?}_{\alpha}(x) $$
for $\alpha \in (0,1)$?
...
- 427
4
votes
1
answer
4k
views
Exponential series is cosh(x), how to show using summation?
I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} \frac{(x)^{2n}}{(2n)!}
$$
I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that $$...
- 635