Prove that $\sum_{j=1}^{\infty} \left[ \frac{n}{2^j}+\frac{1}{2} \right] = n$ for interger $n\ge 1$
66568
Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$
150642
How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?
136066
Find the sum $\sum _{n=1}^{\infty }\left(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n}\right)$
102531
Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-
155005
How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$
85626
prove $\sum_{i=1}^{n}(x_i-\bar{x})^2\lt\sum_{i=1}^{n}(x_i-a)^2$
139229
How to prove by arithmetical means that $\sum\limits_{k=1}^\infty \frac{((k-1)!)^2}{(2k)!} =\frac{1}{3}\sum\limits_{k=1}^{\infty}\frac{1}{k^{2}}$
128913
A proof of $\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+....+\frac{1}{n}$
71162
Showing that $\sum_{n=1}^{\infty} \frac{z^n}{1+z^{2n}}$ converges
15164
Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
125701
Prove that $-2\log(2) = -2 + \sum_{n=1}^{\infty}\frac{1}{n(2n+1)}$
45440
$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$ c0nvergence
104853
Proving that $\lim\limits_{n\to\infty}n\left( \int_0^1 f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\left(\frac k n\right) \right)=\frac{f(1)-f(0)}{2}$
48642
Let $f(n) = \sum_{(k,n)=1} e^\frac{2\pi ik}{n}$. Show that $f(n) = \sum_{k=1}^{n}\left(\sum_{d\mid (k,n)} \mu(d)e^{\frac{2\pi ik}{n}} \right) $
119987
Show the series $ \sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j} $ converges
73865
Find $\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)$
140614
Show that $\sum^\infty_{n=m}\frac{1}{n^2} = O(\frac{1}{m})$