Find $\lim_{n\to \infty} \sum_{k=1}^n \frac{k}{n^2+k^2}$   Since $\frac{k}{n^2+k^2}\leq \frac{k}{k^2+k^2}=\frac{1}{2k}$, then $\sum_{k=1}^n \frac{k}{n^2+k^2}\leq \sum_{k=1}^n \frac{1}{2k}=\frac{1}{2} \sum_{k=1}^n\frac{1}{k}$.  Now we send $n$ to infinity, then since $\sum_{k=1}^\infty \frac{1}{k}$ is harmonic, $\lim_{n\to \infty} \sum_{k=1}^n \frac{k}{n^2+k^2}$ doesn't exist.  I wonder if my thinking is right.
31748
How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$   is unable to calculate it.  This is a question from a questions booklet, and the options for the answer are--  $\begin{align} &amp;A) \dfrac{\pi}{2} \\ &amp;B) \log 2 \\ &amp;C) \dfrac{\pi}{4} \\ &amp;D) \dfrac{1}{2} \log 2 \end{align}$
72225
I have to compute the limit of this sequence ${\textstyle\sum_{k=1}^n}\frac1{\sqrt{n^2+k}}$ as $n\rightarrow\infty$. First I was thinking about some Riemann sum and and forced the $n^{2}$ outside the square root but the function was not so pleasant.
70232
I'm asked to find $\displaystyle\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$.  It seems that the sum converges.  I managed to prove that $\forall n,1&lt;\sum_{k=1}^n \left(\frac{k}{n}\right)^n&lt;e$  But I don't know how to prove convergence, nor find the value of the limit.
42353
I’m having a problem approaching the question. The fact that it’s a limit of a sum is confusing to me.   $$\lim _{n\to \infty }\left(\frac{1}{n^2+1}+\frac{1}{n^2+2}+...+\frac{1}{n^2+n}\right)$$
103880
Just checking. $2^n$ ($n \to \infty$) tends to $\infty$. + $3^n$ also ($n \to \infty$) tends to $\infty$  so the sum gets me $\infty$.  Now $(\infty)^{1/\infty}$ : $(\infty)^0 = 1$  I see no other way.  Theorem: $n^{\frac{1}{n}} = 1$ as $n \to \infty$.  Analogous:  $(2^n - 3^n)^\frac{1}{n}$ = $(\infty  - \infty)^0$ = 1 ???  Is it ok to suppose infinity on any integer $k^n$ as $n$ goes to infinity ?
121100
While Solving some integral problem, I encountered the following infinite series:  $$\displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{n^2k^2(n+k)^2}$$  I have tried many methods including partial fractions... I seek help! Please provide hints if you don't have the complete answer.
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?
21884
I am attempting to prove if the below expression converges or diverges:  $$\sum_{n=2}^{\infty} \frac{1}{n (\ln n)^2}$$  I decided to try using the . So then, I set $f(n) = \frac{1}{n (\ln n)^2}$ and $g(n) = \frac{1}{n}$ and did the following:  $$ \lim_{n \to \infty} \frac{f'(n)}{g'(n)} =  \lim_{n \to \infty} \frac{-\frac{\ln(n) + 2}{n^2 (\ln n)^3}}{-\frac{1}{n^2}} =  \lim_{n \to \infty} \frac{\ln(n) + 2}{n^2 (\ln n)^3} \cdot n^2 =  \lim_{n \to \infty} \frac{1}{(\ln n)^2} + \frac{2}{(\ln n)^3} = 0 $$  I know that $\sum \frac{1}{n}$ diverges due to properties of harmonic series, and so concluded that my first expression $\frac{1}{n (\ln n)^2}$ also must diverge.  However, according to , and as suggested by , the expression actually converges.  What did I do wrong?
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
9339
What is the sum of the 'second half' of the harmonic series?  $$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$  More precisely, what is the limit of the above sequence of partial sums?
90012
I am studying for a test and I am given this problem:  $$\lim_{n\to\infty}\frac{n^n}{n!}.$$  How do I go about solving this limit? Intuitively I see how the numerator is growing much faster, but how do I express this precisely? Thanks!
25829
Show that $$\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{n}{n^2+k^2} = \frac{\pi}{4}.$$  I am familiar with Taylor series and Fourier series of the standard functions. I tried to compare with those and see if there is a relation, I don't seen to find any. How do I tackle this?
89503
Does the series $\sum \limits_{n=1}^{\infty} \sin(n^2)$ converge?  I think I've tried everything, I have no more ideas.
39958
I have this problem :  $$S_n=\sum_{k=2}^\infty \frac{1}{k^2-1}$$  My solution  $$S_n=\sum_{k=2}^\infty \frac{1}{k^2-1} = -\frac{1}{2}\sum_{k=1}^\infty \frac{1}{k+1} -\frac{1}{k-1} = -\frac{1}{2}[(\frac{1}{3}-1)+(\frac{1}{4}-\frac{1}{2})+(\frac{1}{5}-\frac{1}{3})+(\frac{1}{6}-\frac{1}{4})+...]$$  I think that the sum should be $\frac{1}{2}$ since the limit of :  $$-\frac{1}{2}[-1+\frac{1}{k}+...+\frac{1}{n}] = \frac{1}{2}$$  But that wrong. Any ideas?
76156
I need help with the following limit $$\lim_{n\to\infty}\sum_{k=1}^n \frac{1}{\sqrt{kn}}$$  Thanks.
51322
Could please give me a hint.  $$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{k}{k^{2}+1}$$  It looks like an integral, but I failed to figure out the function.
79
How to prove $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty.$$  I try to do like $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=\sum_{N=1}^\infty \sum_{n+m=N}^\infty  \frac{1}{m^2+n^2}=\sum_{N=1}^\infty \sum_{m=1}^{N-1}  \frac{1}{m^2+(N-m)^2}$$  $$\frac{1}{m^2+(N-m)^2}\leq \frac{2}{N^2}$$ but it doesn't work.