All Questions
104
questions
3
votes
3
answers
1k
views
Determining the convergence of $ \sum\limits_{n=1}^{\infty}\frac{-(n-1)}{n\sqrt{n+1}}$
We have to determin the convergence of the following sum:
\begin{equation}
\sum\limits_{n=1}^{\infty}\frac{-(n-1)}{n\sqrt{n+1}}
\end{equation}
And we've seen multiple ways to determine if the sum ...
- 438
3
votes
1
answer
88
views
If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$ then $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$.
Suppose $\{f_{n}\}_{n=1}^{\infty}$ be functions such that $f_{n} : \Bbb{N} \rightarrow \Bbb{R}^{+}$ for each $n$.
I was trying to prove -
If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \...
- 5,032
0
votes
2
answers
102
views
A Closed form for the $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3}$
I'm looking for a closed form for this sequence,
$$\sum_{n=1}^{\infty}\left(\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3} \right)$$
I applied convergence test. The series converges.I want to know ...
user548054
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
0
votes
1
answer
39
views
What does the following sum converge to? is there a closed-form formula?
Is there a closed-form term for the following sum:
$$\sum_{i=z}^{\infty}y^{i}\frac{1-(\frac{x}{y})^{(i+1)}}{1-(\frac{x}{y})},$$
where $x<y<1$ and $z$ is integer grater than $0$.
- 187
1
vote
3
answers
486
views
Implicit definition of recursive sequence $\displaystyle a_{n+1} = a_n + \frac{1}{(3 + (-1)^n)^n}$
I have to show that the recursive sequence given by
$$
b_1 = 1, ~ b_{n+1} = b_n + \frac{1}{(3 + (-1)^n)^n}
$$
converges.
I can show its convergence by showing its monotone and bounded by 2 but I ...
- 4,878
3
votes
2
answers
898
views
Applying the ratio test and uniform convergence
I have been trying to apply the ratio test onto $\dfrac{z^n}{1+z^n}$. After the usual initial steps.
I need to show that $$\lim_{n \to \infty} \left|\dfrac{z(1+z^n)}{1+z^{n+1}}\right|<1$$
I am ...
- 109
3
votes
4
answers
2k
views
Determining whether the series: $\sum_{n=1}^{\infty} \tan\left(\frac{1}{n}\right) $ converges
I was tasked with determining whether the following series:
$$\sum_{n=1}^{\infty} \tan\left(\frac{1}{n}\right) $$
converges.
I tried employing the integral test which failed and produced ...
- 3,231
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
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
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
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
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 $\sum_{n=0}^\infty \frac{1}{n^2}$ ...
1
vote
2
answers
124
views
Is it divergent? $\lim_\limits{n\to\infty}\frac{n!^{3}}{(3n)!}$ [closed]
Could you help me with this limit?
$\lim_\limits{n\to\infty}\frac{n!^{3}}{(3n)!}$
I know that it is divergent. But I don't know to which functions to compare.
- 656