All Questions
104
questions
15
votes
2
answers
331
views
If $\,a_n\searrow 0\,$ and $\,\sum_{n=1}^\infty a_n<\infty,\,$ does this imply that $\,n\log n\, a_n\to 0$?
A quite elegant and classic exercise of Calculus (in infinite series) is the following:
If the non-negative sequence $\{a_n\}$ is decreasing and $\sum_{n=1}^\infty a_n<\infty$, then $na_n\to 0$.
...
- 84.4k
10
votes
1
answer
382
views
On convergence of series of the generalized mean $\sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s.$
Assume that $a_n>0$ such that $\sum_{n=1}^{\infty}a_n $ converges.
Question: For what values of $s\in \Bbb R$ does the following series :
$$ I_s= \sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^...
- 24.2k
8
votes
1
answer
253
views
Proving that $\sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s$ converges when $\sum_{n=1}^{\infty}a_n $ converges
Assume that $a_n\ge0$ such that $\sum_{n=1}^{\infty}a_n $ converges, then
show that for every $s>1$ the following series converges too:
$$\sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\...
- 24.2k
8
votes
2
answers
279
views
What number does $\sum_{k=1}^{\infty}\ln^k(1+\frac{1}{k})$ converge to?
What number does$$\sum_{k=1}^{\infty}\ln^k(1+\frac{1}{k})$$converge to? I think it converges by root test$$\lim_{k\to\infty}\left(\ln^{k}(1+\frac{1}{k})\right)^{\frac{1}{k}}=\lim_{k\to\infty}\ln(1+\...
- 265
7
votes
1
answer
275
views
Calculating $\sum\limits_{n=1}^\infty\frac{{1}}{n+3^n} $
I was able to prove this sum
$$\sum_{n=1}^\infty\frac{{1}}{n+3^n}$$
is convergent through the comparison test but I don't get how to find its sum.
- 89
6
votes
2
answers
209
views
Prove that sum is convergent
How to prove that the following sum is convergent? $$\sum_1^\infty\frac{\sin(n + \ln{n})}{n}$$
I tried to use formula $$\sin(n+ \ln{n}) = \sin{n}\cos \ln{n} + \sin \ln{n}\cos{n}$$ and $$\sum_1^N \sin{...
- 563
5
votes
1
answer
409
views
Dini's theorem (specific case)
Note: I asked this question before but it wasn't well written, So I deleted my previous question and re-wrote it.
According to Dini's theorem:
If $X$ is a compact topological space, and $\{ f_n \}$...
- 421
5
votes
2
answers
111
views
Behavior of a sum on the boundary of convergence/divergence
I am seeking the behavior of the sum
$$
\sum_{n = 1}^{\infty}\frac{\ln\left(n\right)}{n}\,z^{n}
$$
as $z \to 1^{-}$. I know that at $z = 1$, it diverges. So, ideally, I would like to know how it ...
- 323
5
votes
2
answers
346
views
Sum $\sum\limits_{n,m=1}^\infty \frac{1}{(n+m)!},$
I am looking at:
$$\sum_{n,m=1}^\infty \dfrac{1}{(n+m)!},$$
my task is to show that it is absolutely convergent and to find its sum.
I have found the sum doing the following:
$$\sum_{m,n=1}^\...
- 51
4
votes
5
answers
386
views
Show that $\sum_{n=0}^\infty \frac{1}{n+1} \binom{2n}{n} \frac{1}{2^{2n+1}} = 1.$
Question: Show that
$$\sum_{n=0}^\infty \frac{1}{n+1} \binom{2n}{n} \frac{1}{2^{2n+1}} = 1.$$
From Wolfram alpha, it seems that the equality above is indeed correct.
But I do not know how to ...
- 15.9k
4
votes
5
answers
5k
views
Does $\sum_{n=1}^{\infty}\frac{\cos\left(\frac{n\pi}{2}\right)}{\sqrt{n}}$ converge?
Does the following series converge?
$$\sum_{n=1}^{\infty}\frac{\cos\left({\frac{n\pi}{2}}\right)}{\sqrt{n}}$$
The $\cos$ function:
alternates between (-1) and 1 for every $n$ that is even. (for a ...
- 1,084
4
votes
2
answers
139
views
$\sum_{n=1}^\infty (1 - \cos \frac1n)^\alpha\log n$
How do I find for which values of $\alpha \in \mathbb{R}$ the sum converges?
$$\sum_{n=1}^\infty (1 - \cos \frac1n)^\alpha\log n$$
I have tried using the following techniques:
Comparison test. Hard ...
- 2,191
4
votes
1
answer
520
views
infinite series with a constant - calculation: $\sum_{x=1}^{\infty}\frac{c}{x(x+1)(x+3)}=1$
In order to continue solving a probability related exercise, I have to extract the value of the constant $c$ from the following: (while $x\in \mathbb{N}$)
$$\sum_{x=1}^{\infty}\frac{c}{x(x+1)(x+3)}=1$$...
- 137
4
votes
2
answers
488
views
Sum $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}$
I have the following infinite sum:
$$
\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}
$$
Because there is a $(-1)^n$ I deduce that it is a alternating series.
Therefore I use the alternating series ...
user137720
4
votes
1
answer
690
views
Formula for nth derivative of partial sum of geometric series.
I am trying to find a formula for either
(1) the $n$th derivative for the following $m$th partial sum:
$$\frac{d^n}{dx^n} \sum_{i=0}^m x^i$$
or (2) the $n$th derivative of the infinite series given by
...
- 6,099