$$\lim_{n\rightarrow\infty}\sum\limits_{k=1}^n \frac{k}{k^2+n^2}$$  I was trying to write it as an integral but failed, how to do it here?
105571
I guess it has something to do with Riemann sums but this is new for me.  $\displaystyle\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$  How do I start?
13215
$$  \lim_{n\to\infty} n^{1/2} \int_{0}^{1} \frac{1}{(1+x^2)^n}\mathrm{d}x=0 $$  Is my answer correct? But I am not sure of method by which I have done.
87187
I was trying to do this limit $$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$$  and I cant, I will be pleased if someone give me a hint to do it.
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.
13847
I am trying to compute this integral/limit, I don't feel like I have any good insight...   $$\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx.$$  I have tried to make a change of variable to get rid of the $n^2$, I changed to $X=\frac{x}{n^2}$ but got something even worse, I've tried to reach a situation where I could use a convergence theorem for Lebesgue Integrals,... I'm not sure I'm even on the right track!  Could you give me a hint on how to start this?  Thank you very much!
66506
How can I calculate $\sum\limits_{n=1}^{\infty}\frac{1}{F_n}$, where $F_0=0$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$?  Empirically, the result is around $3.35988566$.  Is there a "more mathematical way" to express this?
95733
Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$  I use software to complete the  series is  $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$  I have no idea about it.  :|
77797
Compute $$\displaystyle \lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \sum_{k=1}^n (\sin{x})^kdx.$$   I tried writing $\displaystyle \lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \sum_{k=1}^n (\sin{x})^kdx = \lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \frac{1-(\sin{x})^{n+1}}{1-\sin{x}}dx$, but I don't know how to continue form here.  I know that the limit diverges to $\infty$, but I don't know how to prove it.
34840
Use Riemann integral to evaluate the limit $$\lim_{n\to\infty}\frac{\sum_{k=1}^n \sqrt k}{\sum_{k=1}^n\sqrt{n+k}}$$
30804
I need to show the following: $$\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$$  For all complex numbers z. I don't know how to start this. Should I use l'Hopitals rule somehow?
98362
I am trying to show $$\lim_{n\rightarrow \infty} \int_{[0,1]^n} \frac{|x|}{\sqrt{n}}dx=\frac{1}{\sqrt{3}}$$ where $[0,1]^n$ is the unit cube in $\mathbb{R}^n$. I would appreciate any hint.
19745
Prove that : $\displaystyle \lim_{n\to \infty} \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{2n}=\ln 2$    the only thing I could think of is that it can be written like this :  $$ \lim_{n\to \infty} \sum_{k=1}^n \frac{1}{k+n} =\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{1}{\frac{k}{n}+1}=\int_0^1 \frac{1}{x+1} \ \mathrm{d}x=\ln 2$$  is my answer right ? and are there any other method ?(I'm sure there are)
114239
Find the value of $$ \lim_{n\to\infty} \sum_{k=n+1}^{2n} \frac{1}{k} $$ I think it become a series with each denominator approach infinity. so the value of the limit is 0. am i true ?
58267
How to compute this limit:  $$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$  Please give me some hint.
24341
Evaluate $\sum_{k=1}^{\infty} \frac{1}{k^2(k+1)^2}$.   I've tried to use a telescoping approach, but it doesn't seem to work. It doesn't also seem to converge to a "pretty" value as well.
32091
We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$  What is the limit of this function as $n \rightarrow \infty$?  My idea:  $$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + \dfrac{n}{n^2} = 0$$  Is this correct?
103990
How to find $$\lim_{n \rightarrow \infty}\; \frac{1}{n}\;\sum_{k=1}^{\Big\lfloor\frac{n}{2}\Big\rfloor} \cos\Big(\frac{k\pi}{n}\Big)$$?  I know the method when the upper limit is simply $n$, namely it converges to $\int_0^1 f(x)\;dx$ where $f$ is monotonically increasing on an interval ( in this case our term is of the form $\frac{1}{n} \sum f(k/n)$ ) But here the upper limit is $\Big\lfloor\frac{n}{2}\Big\rfloor$. How to approach this?