All Questions
Tagged with summation power-series
362
questions
2
votes
1
answer
248
views
What are the conditions for Ramanujan's Master Theorem to hold?
Ramanujan's Master Theorem states that if
$$f(x) = \sum_{k=0}^{\infty} \frac{\phi(k)}{k!}(-x)^k$$
then $$\int_{0}^{\infty}x^{s-1}f(x)\ dx = \Gamma(s)\phi(-s).$$
But there are obviously some conditions ...
- 359
0
votes
0
answers
23
views
Summation of the following form with non-integer n
I have the following function:
$$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$
If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be ...
- 109
2
votes
3
answers
110
views
How can I derive the first two terms of the asymptotic expansion of $f(n)=\sum_{k=1}^\infty [(-1)^k/k]\ln(n^2+k)$ at $n \to +\infty$?
I am struggling with the problem in the title of this post. I have tried many different methods, but nothing has worked so far. I only managed to derive the first term of the asymptotic expansion:
$f(...
- 135
1
vote
0
answers
44
views
Multiplication of multiple summations of complex functions
I have a series that looks like $\sum_{l,m,n}\frac{A^{l}B^{m}C^{n}}{l!m!n!}$ where $A$ is a complex function and $B$ and $C$ are real functions. The summation is finite up to some cutoff $p$. $A$, $B$,...
- 21
0
votes
0
answers
39
views
Converting a power series with recursively related coefficients into a single sigma sum expression
EDIT: Ok, silly me. There is an obvious closed form summation which somehow escaped me. Nonetheless, I would appreciate comments on deriving a characteristic polynomial from the generating function.
...
- 434
0
votes
0
answers
60
views
Rewriting a sum with a floor function as upper limit
I am having some trouble in rewriting a sum whose upper limit is given in terms of a floor function $\lfloor \cdot \rfloor$. The task is to prove that both sides of the following expression coincide:
$...
- 91
4
votes
2
answers
223
views
Methods for finding and guessing closed forms of infinite series
I want to prove $\displaystyle\sum_{k \ge 0} \Big(\frac{1}{3k+1} - \frac{1}{3k+2}\Big) = \frac{\pi}{\sqrt{27}}$
The reason for this question is I was doing the integral $\displaystyle\int_0^{\infty} \...
- 1,863
4
votes
2
answers
133
views
Interchanging summations with complicated, nested indices
I have a question regarding interchanging the order of three nested summations. My expression looks like
\begin{align}
\sum_{n=0}^\infty \sum_{k=0}^n \sum_{\nu=0}^{4n-2k}\frac{C_{nk\nu}}{k!(n-k)!}\...
- 91
0
votes
0
answers
48
views
Multiplication of a power series and a finite-order polynomial [duplicate]
I am trying to find a general expression for the coefficients of the power series that results from the multiplication of a polynomial and a power series. I have looked at this post Convolution and ...
- 317
0
votes
1
answer
145
views
Solving a sum similar to geometric series
How do I solve the sum
$$\sum_{k=1}^y \left( 1-\frac{1}{\ln x} \right)^k \hspace{0.5cm} $$
for $x>0$ and $y$ a positive integer greater than one?
Despite resembling a geometric series, it does not ...
user1089309
2
votes
1
answer
167
views
Find the summation of $\sum_{n\geq1}\frac{3^n}{n\left(\frac{1}{n}+1\right)^n}x^n$
I was trying to find what the summation of $$\sum_{n\geq1}\frac{3^n}{n\left(\frac{1}{n}+1\right)^n}x^n$$ is, but I'm kind of stuck.
I recognized the pattern at the bottom as
$$\lim_{n\to+\infty}\left(...
- 41
0
votes
1
answer
76
views
Prove the formula $1+r\cdot \cos(α)+r^{2}\cos(2α)+\cdots+r^{n}\cos(nα)=\dfrac{r^{n+2}\cos(nα)-r^{n+1}\cos[(n+1)α]-r\cosα+1}{r^{2}-2r\cdot \cos(α)+1}$
For $r,a\in\mathbb{R}:\; r^{2}-2r\cos{a}+1\neq 0$ prove the formula $$1+r\cdot \cos(a)+r^{2}\cos(2a)+\cdots+r^{n}\cos(na)=\dfrac{r^{n+2}\cos(na)-r^{n+1}\cos[(n+1)a]-r\cdot \cos(a)+1}{r^{2}-2r\cdot \...
- 21
1
vote
1
answer
93
views
Expanding denominator in a power series, mismatch of the expansion
Below is a snippet from the book Ralston:First course in numerical analysis but it seems to me that something is wrong with $(10.2-11):$ the denominator divided by $a_1\lambda_1^m$ starts with $1$ not ...
- 3,978
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
6
votes
2
answers
115
views
Proving that the exponential satisfies the following sum equation
I was thinking about how $(\sum_{n=0}^{\infty} \frac{1}{n!})^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
$
and was wondering if there existed any other sequences that satisfied this besides the exponential....
- 506