All Questions
41
questions
1
vote
1
answer
60
views
Why does $f^{(k)}(0)$ exists in Rudin's PMA Corollary 8.1?
is plugging $0$ in (6) result to $0^0$?
here is conditions of $8.1$
4
votes
1
answer
88
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.
$$
\lim_{N\rightarrow \infty} \sum_{n=0}^{N-2}{...
- 1,240
0
votes
0
answers
60
views
Rewriting a sum with a floor function as upper limit
I am having some trouble in rewriting a sum whose upper limit is given in terms of a floor function $\lfloor \cdot \rfloor$. The task is to prove that both sides of the following expression coincide:
$...
- 91
0
votes
0
answers
105
views
Manipulation with the following infinite sum
Calculating some observable, I obtained the following-like converges sum
$$
S = \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum_{p=0}^{\min(n,k)} \frac{x^n}{n!} \frac{y^k}{k!} F(p),
$$
where $F$ - some ...
- 121
0
votes
0
answers
65
views
How to find the sum of a power series without knowing the actual power series
How do I find the sum of this series?
$$ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)2^{n-1}} $$
The approach I wanted to use is to find a power series that can become this number series for a certain ...
- 11
0
votes
1
answer
71
views
Evaluate $\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$
Evaluate $$\sum_{i=2}^{\infty}{\frac{n}{n^2-1}}x^n$$ using the fact that
$${\frac{n}{n^2-1}} = {\frac{1}{2(n-1)}} + {\frac{1}{2(n+1)}}$$
So far I have proven that the Radius of Convergence is 1 and ...
- 63
0
votes
2
answers
52
views
What is the Radius of Convergene of $f(z) = \sum_{n=0}^\infty \frac1{4^n}z^{2n+1}$
Here's what I have:
$f(z) = z + \frac14z^3+\frac1{4^2}z^5+...$
So, my coefficients are either $0$ or $\frac1{4^n}$ with $\frac1{4^n}$ being the supremum.
So, $\limsup \limits_{n \to \infty} |c_n|^{\...
- 70
10
votes
2
answers
2k
views
Find the sum: $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$ [duplicate]
Find the sum:
$$\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$$
My try:
I played a bit with the coefficient to make it look easier/familiar:
First attempt:
$$\begin{align}
\sum_{n=0}^\infty \frac{(n!)^2}{...
- 877
1
vote
1
answer
49
views
Show that $F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$
I've been working on a recent exercise question where I was asked to show that:
$$F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$$
Now I cansee that the infinite sum is ...
- 115
1
vote
4
answers
109
views
Evaluating the sum $\sum\limits_{n=1}^{\infty}\frac{x^{2n}}{(2n+1)!}$
I'm having difficulties with the sum above. My first attempt was to rewrite it like
$$
\sum\limits_{n=1}^{\infty}\frac{x^{2n}}{(2n+1)!}=\frac{1}{x}\sum\limits_{n=1}^{\infty}\frac{x^{2n+1}}{(2n+1)!}
$$
...
- 81
0
votes
1
answer
46
views
Sum of a finite series almost like gp
Let $a>1$ and consider the following finite series:
$$
1+\frac{2}{a}+\frac{3}{a^2}+\cdots+\frac{n}{a^{n-1}},
$$
where $n\geq 1$ is a fixed quantity.
Then is the above series uniformly bounded by a ...
- 713
1
vote
3
answers
117
views
Suppose $\sum_{n=1}^{\infty}a_nx^n=f(x)$, then what can we say $\sum_{n=1}^{\infty}\frac{a_n}{n^3}x^n$ and the limit?
Let $f(x)=\frac{1}{(1-x)(1-x^4)}$, let $a_n$ be the nth term of the maclaurin expansion of $f(x)$.
What can we say about the power series $a_0+\sum_{n=1}^{\infty}\frac{a_n}{n^3}x^n$? Can we express ...
- 293
0
votes
0
answers
144
views
Radius of Convergence of the 'product' of two power series of the same radius of convergence
I have come across a problem involving the radius of convergence of power series that I can't solve for the life of me, any help would be greatly appreciated. I cannot use the lim sup method to solve ...
- 41
2
votes
2
answers
138
views
Finding the exact sum of this power series
I am currently studying power series and have come across a problem I am having difficulties with. I have done some looking around on the website for a similar problem but I cant find anything that ...
- 41
0
votes
2
answers
253
views
Infinite power series sum $\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)3^n}$
Using theorems about differentiation or integration of power series calculate infinite sum of
$$
\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)3^n}
$$
The answer should equal to $\frac{\pi}{2\sqrt3}$.
I ...
- 2,190