All Questions
13
questions
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
3
votes
4
answers
654
views
Prove that $\sum_{n=1}^{\infty}\exp\left(-n^{\varepsilon}\right)$ converges.
I have the problem to prove, that the following series converges:
\begin{equation}
\sum_{n=1}^{\infty}\exp\left(-n^{\varepsilon}\right),
\end{equation}
where $\varepsilon > 0$.
I tried everything....
user546010
2
votes
3
answers
103
views
$\sum_{n=1}^{\infty} u_n^2=0$ $\Rightarrow$ $u_n=0 \ \forall \ n\in \mathbb{N}$
Let, $<u_n>$ be a real sequence and given that $$\sum_{n=1}^{\infty} u_n^2=0$$
Prove that $u_n=0 \ \forall \ n\in \mathbb{N}$
Attempt
$u_n^2\geq 0 \ \forall \ n\geq 1$
Since, $$\sum_{n=1}^{\...
user908035
1
vote
2
answers
3k
views
Does a series converge if its initial value is undefined?
I understand that the series $\sum_{n=1}^\infty \frac{1}{n^2}$ converges.
What happens when you start at a different value than n=1?
For example, does the series $\sum_{n=0}^\infty \frac{1}{n^2}$ ...
4
votes
5
answers
386
views
Show that $\sum_{n=0}^\infty \frac{1}{n+1} \binom{2n}{n} \frac{1}{2^{2n+1}} = 1.$
Question: Show that
$$\sum_{n=0}^\infty \frac{1}{n+1} \binom{2n}{n} \frac{1}{2^{2n+1}} = 1.$$
From Wolfram alpha, it seems that the equality above is indeed correct.
But I do not know how to ...
- 15.9k
3
votes
4
answers
29k
views
Is $\sum\frac{1}{\sqrt{n+1}}$ convergent or divergent?
$$\sum\frac{(-1)^n}{\sqrt{n+1}} \text{and} \sum\frac{1}{\sqrt{n+1}}$$
The first one is an alternating series, so it would just be:
$$\sum (-1)^n\frac{1}{\sqrt{n+1}}\Rightarrow \;^\lim_{n\rightarrow\...
- 381
3
votes
2
answers
154
views
Convergence of $S_p = \sum_{k=1}^{p}\left( \sum_{j=1}^{k}\frac{r^{k-j}}{j}\right)^2$
I'm looking at the following sum
$$ S_p = \sum_{k=1}^{p}\left( \sum_{j=1}^{k}\frac{r^{k-j}}{j}\right)^2$$
where $r \in (0,1), p > 1$. The goal is to inspect whether the sum converges to some ...
- 740
3
votes
3
answers
81
views
Does $a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... + \frac{1}{\sqrt{n^2+2n-1}}$ converge?
$a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... + \frac{1}{\sqrt{n^2+2n-1}}$
and I need to check whether this sequence converges to a limit without finding the limit itself. I think ...
- 31
2
votes
2
answers
367
views
Cauchy product summation converges
I had a previous question here, which I'm quoting:
How can I prove that the following summation converges?
$$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1) (n-k+1)}$$
I tried to prove ...
- 421
2
votes
1
answer
598
views
Rearranging a series of nonnegative terms
Let $a_1, a_2, a_3,...$ be a sequence of non-negative real numbers, let $S_1, S_2, S_3,...$ be a sequence (finite or infinite) of disjoint nonempty sets of natural numbers whose union is $\{1,2,3,...\}...
0
votes
2
answers
71
views
How come it be $\frac{3}{2}A$ and not only $A$?
OK I admit I was too lazy to type this question so I took a screenshot , I got it from the site @brilliant.org where it asked in terms of $A$ what would be the 2nd summation equation ? The explained ...
- 763
0
votes
2
answers
37
views
Summation with two costs
I need to find for which alpha and beta values, the following summation will converge and for which it will diverge
$$\sum\limits_{n=1}^\infty (-1)^{n-1}\left(\alpha-\frac{(n-\beta)^n}{n^n}\right)$$
...
- 421