All Questions
Tagged with summation power-series
362
questions
1
vote
1
answer
99
views
Calculating the sum of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$ [duplicate]
I want to find the sum of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$.
I have tried to turn it into a power series for a known function, with no luck. I also tried to write it as $\sum_{n=0}^{\infty} \...
- 211k
4
votes
2
answers
241
views
General formula for the power sum $\sum_{k=1}^{n}k^mx^k\quad \forall\; m\in \mathbb{N}$
In my last question, it turns out to be solving the formula of $\sum_{k=1}^{n}k\omega^k$. I am curious if there is a geranal formula for the power sum:
$$\sum_{k=1}^{n}k^mx^k\quad \forall\; m\in \...
- 482
0
votes
2
answers
119
views
How do I find the partial sum of the Maclaurin series for $e^x$?
In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$.
Wolfram | ...
- 2,269
1
vote
1
answer
99
views
Evaluate the sum of $1/n^6$ using Euler's method
I've just learned how Euler evaluated $1+1/2^2+1/3^2+...+1/n^2+...=\pi^2/6$ by comparing the coefficients of the series form and product form of $\sin(x)/x$.
The series form is
$$\dfrac{\sin(x)}{x}=1-\...
- 3,185
4
votes
1
answer
89
views
A conjecture involving series with zeta function
Recently, I tried to evaluate a limit proposed by MSE user Black Emperor. In the process of evaluating the limit, I have obtained the following equality.
$$
\lim_{N\rightarrow \infty} \sum_{n=0}^{N-2}{...
- 33.2k
2
votes
1
answer
251
views
What are the conditions for Ramanujan's Master Theorem to hold?
Ramanujan's Master Theorem states that if
$$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).$$
But there are obviously some conditions ...
- 5,337
0
votes
0
answers
23
views
Summation of the following form with non-integer n
I have the following function:
$$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$
If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be ...
- 119
2
votes
3
answers
174
views
Sum to infinity series: $\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}$
Consider the following series $$\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}$$
I'm told to analytically find the sum to infinity and I have been given this as a clue.
$$\Sigma_{k=0}^\infty x^k = ...
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(...
- 268k
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:
$...
- 50.3k
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} \...
- 20.7k
4
votes
2
answers
135
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)!}\...
- 109k
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