EDIT: I want to prove it with &quot;converges absolutely&quot; and not something else. I want to prove that the sequance $\sum _{n=1}^{\infty \ }\left[n^3\sin \left(\frac{1}{n}\right)-n^2+\frac{1}{6}\right]$ converges. I thought of trying converges absolutely, but I don't even know how to start. Also, trying to derivate the inside function doesn't help showing that the function is a non-negative one. Any ideas? Thanks!
123580
Let $t\in \mathbb R$. Prove that $\displaystyle{\sum_{n=1}^{\infty}n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+t}$ converges iff $t=\frac{1}{6}$.  While the $(\Rightarrow)$ direction was fairly easy (as $a_n$ should converge to $0$, forcing $t=\frac{1}{6}$), i'm facing some hard time trying to prove the second direction, that is $\displaystyle{\sum_{n=1}^{\infty}(n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+\frac{1}{6})}$ converges. I did notice that $a_n=a_{-n}$ and tried to write this series as a Leibnitz series, but had no luck since the sequence doesn't decrease. I also tried to prove absolute convergence but then again, no luck. Thanks.
103129
I have a function defined as: $$S(a)=\sum_{n=1}^\infty\frac{\sin(n)}{n^a}$$ My question is for what values of $a$ is this convergent, and how can I evaulate this?  For starters, I know that $S(a)$ is definately convergent  for a $a\ge2$ since: $$\sum_{n=1}^\infty\frac{\sin(n)}{n^a}\le\sum_{n=1}^\infty\frac{1}{n^a}$$ So we can use $\zeta(a)$ as the upper limit for the series, which makes me think I can extend this convergence to $a&gt;1$. However, I have no idea how I would evaluate this. Thanks!
43842
Let $\sum a_n$ be a series of non-negative terms and let $$L = \lim_{n\to\infty}n\left(1-\frac{a_{n+1}}{a_n}\right)$$ Prove that the series converges (resp. diverges) if $L &gt; 1$ (resp. $L&lt;1$). I've tried, for example, that when $L&lt;1$, $$n\left(1-\frac{a_{n+1}}{a_n}-L\right)= \frac{n(a_n-a_{n+1}-La_n)}{a_n}\ge\frac{n(a_n-a_{n+1}-a_n)}{a_n}=\frac{-na_{n+1}}{a_n}$$ and then using epsilons and the sort, but I can't get anywhere. Any tips?  P.S. using Kummer's test doesn't count
34891
I need to find out whether the following series converges or diverges. If it converges I also have to discover whether it is an absolute convergence or a conditional one. $$\sum_{k=1}^{\infty}\left(\frac{1+\cos k}{2+\cos k}\right)^{2k}$$ Root test did not help me much since $\lim_{k\rightarrow\infty}\left(\frac{1+\cos k}{2+\cos k}\right)^2$ may equal anything from $0$ to $4$. Comparison tests also did not work well. I guess this series is convergent, though. But I still have no idea how to prove it.
10995
How do I find:  $$\prod_{n=1}^\infty\; \left(1+ \frac{1}{\pi ^2n^2}\right) \quad$$  I am pretty sure that the infinite product converges, but if it doesn't please let me know if I have made an error. Also, could I have a nice explanation as to how would someone arrive to the answer.  Thanks alot.
57252
How to prove that the series $$\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$$ is convergent? What about finding the sum?  My attempt: $$\ln (1-1/n^2)= \ln(n-1) -2\ln n + \ln(n+1)$$ The term in the log is decreasing so I get the feeling that the series converges but confused how to prove this and also find the sum.
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.
107313
Let $x_1=1$ and $$x_{n+1} = \frac12\left(x_n + \frac2{x_n}\right).$$ Prove or disprove $(x_n)$ is convergent and show the limit.   When I tried working on it I found the sequence was bounded by square root of 2 and it is was monotone. But apparently the sequence is not bounded by square root of two and is not monotone. But I have no idea why. Any help would be greatly appreciated! Thanks!
1102
I need to prove the following:  $$\lim_{s \to \infty} \sum_{n = 1}^\infty \frac 1 {n^s} = 1$$  This is my attempt:  \begin{align} \lim_{s \to \infty} \sum_{n = 1}^\infty \frac 1 {n^s} &amp; =  \lim_{s \to \infty} \left( 1 + \sum_{n = 2}^\infty \frac 1 {n^s} \right) \\ &amp; = 1 + \sum_{n = 2}^\infty \lim_{s \to \infty} \frac 1 {n^s} \\ &amp; =  1 + \sum_{n = 2}^\infty 0 \\ &amp; = 1 \end{align}  I need help justifying the interchange of the sum and the limit though. Is it enough to show the limit exists and the sum is convergent? I read that I need to prove uniform convergence, how can I do this?
115877
Test for convergence  $$\sum_{n=1}^{\infty} \frac{1}{n^n}$$  I'm at a loss on what to do, is this a geometric series $\frac{1^n}{n^n}$?
31118
If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n)(b_n)$ converge?  Could someone help me to solve this or at least give me a hint?, I have tried using Cauchy's criterion, the Dirichlet test for convergence, etc, but I can´t prove it.Honestly I don´t know where to start. Any help will be appreciated.
96012
Possible Duplicate:          So, in my calculus class (one I'm teaching, not taking), the sum $\sum_{n=1}^\infty \frac{\sin(n)}{n}$ has come up a few times.  Unfortunately, as someone not exactly an expert in the convergence of sums, it seems to resist the few techniques I know.  Certain none of the usual first year calculus tricks (integral test, alternating series test, ratio test, etc.) work, and the only more tricky technique, partial summation, I can think of doesn't seem to work either (one would need that $\sum_{n=1}^N\sin(n)$ is bounded, which I believe is false).  It seems like it should converge, since it switches sign quite often, but on the other hand, the harmonic series can mess with your intuition, so I don't have much trust in that.  So, I ask to you:     Does this series converge?
56039
I think that the series $$\sum_{n=1}^{\infty} \dfrac{\sin(n)}{n}$$ converges conditionally, but I´m not able to prove it. Any suggestions ?
33274
The sum is $$\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$$ Is this convergent or divergent? I tried to use the divergent test but the test fail because $a_n = (n+2^n)/(n+3^n) = 0 $ as $n$ goes to infinity. Could someone point me to the right direction?  Thanks
9865
I think the radius of convergence for  $\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)x^n$, $x\in \mathbb R$ is:   $r^{-1}$=$\lim_{n\to \infty}$$|\frac{a_{n+1}}{a_n}$|=1   so we get that $r$=1.  But how can I show it formally?  After that, I have to show that $\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)x^n$ is uniformly convergent in the interval $[-r,r]$. Can I maybe use Weierstrass majoranttest? What can I use as majorant serie?
74043
Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$   converge?   I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in advance.
46234
Let ${a_n}$ and ${b_n}$ sequences with positive terms, such that $\sum a_n$ and $\sum b_n$ converges. Prove that $\sum a_nb_n$ converges as well.  What I did: $\sum a_nb_n \le \left(\sum a_n\right) \left(\sum b_n\right)$ and by comparison test we are done.  However, when I asked my teacher regarding that solution, he told me that I can assume that inequality for finite numbers but for infinite numbers, it is not valid. It seems to me that $\left(\sum a_n\right) \left(\sum b_n\right)$ is absolute converges so that are no Riemann issues with this.  Can any one enlighten me and explain why is this invalid?