All Questions
6
questions
11
votes
4
answers
2k
views
A power series $\sum_{n = 0}^\infty a_nx^n$ such that $\sum_{n=0}^\infty a_n= +\infty$ but $\lim_{x \to 1} \sum_{n = 0}^\infty a_nx^n \ne \infty$
Let's consider the power series $\sum_{n = 0}^{\infty} a_nx^n $ with radius of convergence $1$. Moreover let's suppose that : $\sum_{n = 0}^{\infty} a_n= +\infty$. Then I would like to find a sequence ...
- 666
6
votes
1
answer
280
views
Sum of $\sum_{n \geq 1} \frac{(\ln x +1)^n}{n^n}$
I want to find the sum of the following series
$$\sum_{n \geq 1} \frac{(\ln x +1)^n}{n^n}$$
Using theorems on integration and differentiation of series. I can set $t=\ln x+1$ so that I get
$$\sum_{...
- 2,517
4
votes
5
answers
338
views
Power series summation [closed]
Trying to find the sum of the following infinite series:
$$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$
Any ideas on how to find this sum?
- 277
4
votes
4
answers
187
views
What is the sum of $\sum_{k=0}^\infty \frac{(-1)^{k+1}}{2k+1}$?
What is the sum of the following expression:
$$\sum_{k=0}^\infty \frac{(-1)^{k+1}}{2k+1}$$
I know it is convergent but I cannot evaluate its sum.
- 24.5k
3
votes
1
answer
2k
views
Find a sum of a convergent series
Let $x_n$ be a sequence that is given by the following recursive formula:
$x_{n+1} = x_n^2 - x_n +1$, where $x_1=a \gt 1$.
Find: $$\sum_{n=1}^{\infty} \frac{1}{x_n}$$
Not sure really how to ...
- 415
1
vote
1
answer
89
views
How do I evaluate $\sum_{k=1}^nk^pr^k=?$
For this entire post, we have $r\ne1$, $n\in\mathbb N$. For the first half, $p\in\mathbb N$, and at the end $p\in\mathbb Q$.
It is well known that
$$\sum_{k=1}^nr^k=\frac{1-r^{n+1}}{1-r}$$
And
$$\...
