How to find the value this sum converges to?$$\sum_{n=0}^{\infty}n·\frac{3^{n-1}}{4^{3n+1}} $$ I've no clue how to simplify $ \frac{3^{n-1}}{4^{3n+1}} $ so if you could help me with that I could do the rest by myself so any hint would be appreciated.
16813
I would like to know: How come that  $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$  Why isn't it infinity?
56816
$$ \sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$$  Does it converge or diverge?   Can we have a rigorous proof that is not probabilistic?   For reference, this question is supposedly a mix of real analysis and calculus.
11505
Find the values of $p$ for which the series $\sum_{n=2}^\infty \frac{1}{n(\ln n)^p}$ is convergent. I know that the $p$-series $\sum_{n=1}^\infty \frac{1}{n^p}$ is convergent if $p&gt;1$ and divergent if $p\leq1$. So with the series in question, I can't just say &quot;when $p&gt;1$&quot; right? Because the denominator is $n(\ln n)^p$ not just $n^p$ as the formula states. So how should I go about tackling this problem? I know that doing a u-sub for $\ln n$ would work because $du$ would equals $\frac{dn}{n}$ but what would I do with the $p$? Thanks for the help.
75893
This is the sum   $$\sum_{n=1}^{\infty}\frac{1}{{5^n}^!}$$  My first attempt was to assume that the series does converge to a rational number $a/b$. But the $n!$ bothered me and I failed in my proof.  How would you try to prove this series?
64621
Can anyone tell me about the sum of the series $$\sum_{n=1}^\infty \frac{1}{(2n)(2n+1)}?$$ This is not a usual telescoping sum in which all the terms cancel out.
121486
Per the title, does $\displaystyle\sum_{n=1}^{\infty} \sin(\pi(2+\sqrt{3})^n)$ converge? Converge absolutely?   I'm stuck on this question, not sure how to approach it.  Thank you!
17721
How to test whether $\sum_{n=1}^{\infty }\frac{1}{n^{1+\frac{1}{n}}}$ diverges or converges? Can I apply ratio test? Thanks in advance.
39848
Does the following series converge or diverge:  $$\sum_{n=1}^\infty \frac{3^{1/\sqrt{n}}-1}{n}$$  Many thanks!
53582
How can I find the value of the series $$\sum_{n=1}^\infty \frac{1}{(b+n)(a+n)}$$ where $a,b\in[0,1)$?  It is obvious by standard arguments that the series converges, but how can I derive the explicit value dependent on $a$ and $b$.  Thanks  EDIT: $a,b$ can be assumed to be different.
40584
Find this sum     $$I=\sum_{n=0}^{\infty}\dfrac{1}{(3n+1)(3n+2)(3n+3)}$$   My try: let     $$f(x)=\sum_{n=0}^{\infty}\dfrac{x^{3n+3}}{(3n+1)(3n+2)(3n+3)},|x|\le 1$$   then we have $$f^{(3)}(x)=\sum_{n=0}^{\infty}x^{3n}=\dfrac{1}{1-x^3}$$ then we find the $f(x)$,Following is very ugly(can you someone can post your follow solution,) have other simple methods? Thank you very much.
86369
Check whether $$\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\frac{1}{\left(m+n\right)^2}$$ converges or NOT?  My Try:- $\sum _{m=1}^{\infty }\lim_{i\to \infty} \sum _{n=1}^{i}\frac{1}{\left(m+n\right)^2}=\lim_{j\to \infty}\sum _{m=1}^{j }\lim_{i\to \infty} \sum _{n=1}^{i}\frac{1}{\left(m+n\right)^2}=\lim_{j\to \infty}(\lim_{i\to \infty} \sum _{n=1}^{i}\frac{1}{\left(1+n\right)^2}+ \lim_{i\to \infty} \sum _{n=1}^{i}\frac{1}{\left(2+n\right)^2}+ \lim_{i\to \infty} \sum _{n=1}^{i}\frac{1}{\left(3+n\right)^2}+...\lim_{i\to \infty} \sum _{n=1}^{i}\frac{1}{\left(j+n\right)^2})\ge \lim_{j\to \infty}(j\lim_{i\to \infty} \sum _{n=1}^{i}\frac{1}{\left(j+n\right)^2})$  How do I complete the conclusion?
8268
I need to solve    $$\sum_{j = 1}^{\infty} \sqrt{\frac{j!}{j^j}}$$  Does this converge or diverge and why?
113494
Does the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$ converge..? Can anyone please give me a simple proof..
27086
$$\sum_{n=1}^\infty \frac{i^n}{n}$$  I'm not sure how to approach this series to find if it converges/diverges because of the $i^n$. I tried using the comparison test comparing it with $\frac{1}{n}$ and the ratio test, but didn't get anywhere. How would you approach this and look at series of this form or how to look at $i^n$ in an infinite series?
106805
So I just started learning series, and I am having trouble understanding them. I have a series below:  $$\sum_{n=0}^{\infty} \frac{1}{(2n+1)2^n}$$  Can someone explain to me how I should go about this, from start to finish? I couldn't find any online websites that explained how to find sums, so if you have that, it would be greatly appreciated!
123584
Sorry for the oversimplified question, but does the series $\sum_{n=1}^{\infty} \frac{1}{n+1}$ converge? The ratio test of it gives the result of "1". Thanks a lot.
49734
How can I find to what this sum converges to?  $$\sum _{n=1}^{\infty} n(n+1)x^n$$   I proved that it converges when  $$|x| &lt; 1$$  but no idea how to find what it sums to.