The series $$\sum_{k=1}^\infty \frac{1}{k^k}$$ clearly converges. Is it possible to compute its value analytically?
22891
In the solution of the so-called , one of the key steps is to compute $$\int_0^1 x^n (\log x)^n~\mathrm dx$$ using the change of variables $x = \exp\left(-\frac{u}{n+1}\right)$ to obtain the Gamma function.  This substitution to me, looks like it was pulled out of thin air. Can someone help me motivate it? How would I have thought of this substitution?
110985
How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$  I tried to prove that by proving that the following summation in in absolute value converges so the original one converges too, but that's incorrect.  Any other ideas?
48067
I can clearly see that $\dfrac{n!}{n^n}\to 0$ when $n\to\infty$. But how do I know if the sum $$\sum_{n=1}^{\infty}\frac{n!}{n^n}$$ is convergent or not? I know this might be basic, but thank you if anyone can help me.
113494
Does the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$ converge..? Can anyone please give me a simple proof..
25201
My partner is tutoring a Civil Engineering student in maths. Conveniently, I am a civil engineer so when any maths that's confusing comes up, I can usually help out. However, we are having a problem with the summing of power series, one area where I am sorely lacking.  After spending two hours on a question, we still cannot solve it and are always left with a $k$. Admittedly, I don't fully know what I am doing.     Evaluate $\displaystyle \sum_{k=1}^\infty\frac{x^k}{k(k+1)}$.   We do know that the answer is $1 + \dfrac{1-x}{x}\ln (1-x)$.  Any hints or help in the direction to go would be very much appreciated.
2372
Is this series known to converge, and if so, what does it converge to (if known)?  Where $p_n$ is prime number $n$, and $p_1 = 2$,  $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$
30633
Does $\sum_{k=1}^\infty\frac1{k^n}$ converge for $\Re(n)=1,\Im(n)\ne0$?  The ratio test is inconclusive.  It passes the term test for $\Re(n)=1$, but this is not sufficient to prove convergence.  Since we are dealing with so many complex numbers, I do not know of any convergence tests for this.  I know that if it is convergent, it is conditionally convergent.
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.
44494
$$\sum_{n=1}^\infty \frac 1{n!}$$   I think we may compare it with $\frac{1}{n^n}$, and say that $\frac{1}{n!}&gt;\frac{1}{n^n}$. But I am not sure of the convergence or divergence of $\frac{1}{n^n}$ itself, so this turns out to be useless.  What other way is there?
91794
Let  $\sum_{k=1}^{\infty}a_k$ be a series of real numbers. Suppose that $\sum_{k=1}^{\infty}a_kb_k$ converges for every bounded sequence {$b_k$} ? Prove that $\sum_{k=1}^{\infty}a_k$ converges absolutely.  Here is my thought process:  Since {$b_k$} is bounded, every |$b_k$| $\leq$ M for all k.  So $\sum_{k=1}^{\infty}|a_kb_k|$ = $\sum_{k=1}^{\infty}|a_k||b_k|$ $\leq$ $\sum_{k=1}^{\infty}|a_k|M$ = $M$$\sum_{k=1}^{\infty}|a_k$| and since $\sum_{k=1}^{\infty}|a_kb_k|$ converges, $\sum_{k=1}^{\infty}|a_k|$ also converges.  However, I am having a hard time figuring out how to prove absolute convergence. I realize that they are similar questions on this website but they're asking to prove that $\sum_{k=1}^{\infty}a_kb_k$ converges, rather than $\sum_{k=1}^{\infty}a_k$ converging absolutely. Would really love any help, I'm quite stuck!
93967
EDIT:  It is known that  $$ \sum_{k=0}^{\infty} x^k = \frac{1}{1-x} $$  Therefore, is it also possible to simplify  $$\sum_{k=0}^{\infty} x^ky^{\frac{k(k+1)}{2}}$$ into something similar to the infinite geometric series sum?
117700
Determine whether the series $\sum_\limits{k=1}^∞ \frac2{k^{3/2}}$ converges or diverges. If it converges, find the bounds for the sum of the series. A) Converges, $\frac3{2}$ &lt; $\sum_\limits{k=1}^∞ \frac2{k^{3/2}}$ &lt; $\frac5{2}$ B) Converges, 2 &lt; $\sum_\limits{k=1}^∞ \frac2{k^{3/2}}$ &lt; 3 C) Converges, 4 &lt; $\sum_\limits{k=1}^∞ \frac2{k^{3/2}}$ &lt; 6 D) Converges, 6 &lt; $\sum_\limits{k=1}^∞ \frac2{k^{3/2}}$ &lt; 8 E) The series diverges I know the series converges. Using p-Series, p=$\frac3{2}$ meaning that p&gt;1 therefore the series converges. However, I don't know how to find the bounds for the sum of the series
15494
Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.  I've tried applying all the basic tests to no avail. I need to find out what the "trick" for this particular series is.
62310
Let $\sum_{k=1}^\infty a_k$ be a convergent series with nonnegative terms and let $a_{n_k}$ be a subsequence of {$a_k$}. Prove that the series $\sum_{k=1}^\infty a_{n_k}$ converges.  I am having trouble figuring out how to prove this and am pretty lost.  My very first idea was to try and prove that it is increasing but don't know where I would take it from there. Any help would be appreciated!
52961
Does the sequence $$\displaystyle \frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$$   converge?   Attempt. Since $\Big(\frac{k}{k+1}\Big)^k \rightarrow 1/e\neq 0$ and the terms are positive, the series $\sum\limits_{k=1}^{\infty}\Big(\frac{k}{k+1}\Big)^k$ diverges to $+\infty$. I find hard to determine if $n$ or the sum goes faster to $+\infty.$   Thanks in advance.
66479
I can prove this sum has a constant upper bound like this: $$\sum_{k=1}^n \frac1{k ^ 2} \lt 1 + \sum_{k=2}^n \frac 1  {k (k - 1)} = 2 - \frac 1 n \lt 2$$ And computer calculation shows that sum seems to converge to 1.6449.  But I still want to know:  Does this sum converge? Is there a name of this sum (or the series $\frac{1}{k^2} $)?
82381
Is the following series convergent?  $$\sum_{n=1}^{\infty}\frac{e^n\,n!}{n^n}$$  I treid the Ratio and Root tests, but both of them failed.