$$\sum_{n=0}^{\infty } \left ( \frac{1}{n\cdot(n+1)\cdot(n+2)))} \right)$$  I've tried all of the tests but I couldn't solve this.  Does it converges? What is the sum if it converges?
24190
I got this question in my maths paper     Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$   and find the sum if it exists.   I managed to show that the series converges but I was unable to find the sum. Any help/hint will go a long way. Thank you.
95762
As a follow-up to a , I'd like to know if  $$\sum_{n=1}^\infty\frac{1}{2 ^ n}$$  is convergent? If yes, then what does it converge to?  Just as a sidenote - the whole issue started with reading about Zeno's Turtle Paradox which made me curious about the underlying mathematics.
53324
How can I prove that $$\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1 \,\,\, ?$$  I do know a way to prove this (see my answer) but I'm curious to know what other approaches could be taken in dealing with it.
62807
I tried everything that I know but I couldn't solve this series:  $$\sum_{n=0}^{\infty} \left(\frac{\pi}{2} - \arctan(n)\right)$$  Does it diverge or converge?
83494
If $\sum_{n=2}^{\infty}a_n$ converge, then also converge this series? $$\sum_{n=2}^{\infty}\frac{\sqrt{a_{n}}}{\ln\, n}(n^{a_{n}}-1)$$  Please verify my answer below  Counterexample:  $$a_{n}=\begin{cases} \frac{1}{k^{2}} &amp; n=k!^{k^{2}}\\ \\ 0 &amp; \text{All other cases} \end{cases}$$  When our infinites sum is equal to  $$\sum_{k=1}^{\infty}\frac{1}{k}\cdot\frac{1}{2\, \ln\, k}\cdot(k!-1)$$
34609
how can I find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $? It is so weird for me because I put this to Mathematica and it tells me that sum does not converge...  Let consider sum no to infinity, but to n  $$ \sum_{k=1}^{n} (k\cdot \ln \frac{2k+1}{2k-1} - 1) =$$ $$ ln \frac{3}{1}\cdot \left(\frac{5}{3}\right)^2 \cdot...\cdot \left(\frac{2n+1}{2n-1}\right)^n - n = ln \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1} - n $$ but $$ n = ln e^n $$ so it will be $$ln\frac{1}{e^n} \cdot \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1}$$  So the limit of it is $-\infty$ Have I done this well or I missed sth?
13007
What is the sum of $$\sum_{n=0}^{\infty}\frac{1}{2^n(n+1)}$$    I've spent an insane amount of time on this problem. I checked on Wolfram and it gives $ln(4)$, which I assume you get from $2\ln(2)$.   I know that the integral of $\frac{1}{n+1}$ is $\ln(n+1)$. What am I missing?  Thank you!
73322
I'm trying to find the infinite sum that is defined by:  $$ 3 \cdot \frac{9}{11} + 4 \cdot \left(\frac{9}{11}\right)^2 + 5 \cdot \left(\frac{9}{11}\right)^3 + \cdots $$  However, I do not know of any known formula to do this. Am I missing something really simple? Thanks!
34647
Does the following series converge or diverge? I would like to see a demonstration.  $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$  I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} \leqslant \sum_{n=1}^{\infty} \frac{1}{n^{1 + {\sin^{2}}(n)}} \leqslant \sum_{n=1}^{\infty} \frac{1}{n^{1 + {\sin^{2n}}(n)}}. $$
15494
Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.  I've tried applying all the basic tests to no avail. I need to find out what the "trick" for this particular series is.
9714
Find this sum $$\sum_{n=1}^{\infty}\dfrac{1}{n!(n^4+n^2+1)}$$  My try: since $$n^4+n^2+1=(n^2+1)^2-n^2=(n^2+n+1)(n^2-n+1)$$ then $$\sum_{n=1}^{\infty}\dfrac{1}{n!(n^4+n^2+1)}=\sum_{n=1}^{\infty}\dfrac{1}{2n\cdot n!}\left(\dfrac{1}{n(n-1)+1}-\dfrac{1}{n(n+1)+1}\right)$$ then I can't go further.
91006
$$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$$  The series converge?  Please verify my solution below
59280
Sum of $$\lim_{n\rightarrow \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+2}+\cdots \cdots \cdots +\frac{n}{n^2+n}\right)$$  $\bf{My\; Try::}$ I have solved it  using Squeeze theorem  $$\lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{n}{n^2+n}\leq \lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{n}{n^2+r}\leq \lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{n}{n^2+1}$$  So we get $$\lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{n}{n^2+r} = 1$$  My question is can we solve above limit without using Squeeze Theorem,   If yes then plz explian me, Thanks
62940
Assume that $0 &lt; a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that  $$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} &lt; \infty $$ ?  I think it is but I can't prove it. Of course if $a_n \geq \varepsilon$ for some $\varepsilon &gt; 0$ this is obvious.  Any idea? Thanks.
68504
I was doing math and I came across this series     $$\sum_{n=1}^\infty\sin n\cdot\sin\left(\dfrac1n\right)$$   I managed to prove that this series is convergent, but I was wondering what value the series converges to $?$ I've put it on Scientific workplace and got a value of around $0.92 $ But I have no idea how to get there.
21227
Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
19214
Let $\{x_n\}_{n=1}^{\infty}$ be a convergent sequence in a  $(S, ||\cdot ||)$. Find the limit of the sequence \begin{align} \left\{\frac{x_1 + 2x_2 + \cdots + nx_n}{n^2}\right\}_{n=1}^{\infty} . \end{align}  Could you please give me some hints where to start?