Can we prove that $2^{-x}\sum_{k=0}^{\infty} {x \choose k} =1, x \in R^+$
94927
Can we calculate $2^k$ using this easy Taylor series?
125148
Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$
78632
Summation equation for $2^{x-1}$
142057
Prove that if $\sum_{n=1}^\infty a_n$ converges then $\sum_{k=1}^\infty A_k$ converges
72377
Show that $\sum_{n=1}^\infty r^n\cos(n\theta)=\dfrac{r\cos\theta -r^2}{1-2r\cos\theta+r^2}$ whenever $0<r<1$.
144078
What does $\sum_{k=0}^\infty \frac{k}{2^k}$ converge to?
101052
Find the infinite sum $\sum_{k=1}^{\infty}\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2} $
16776
How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$
14487
Does $\sum_{n=2}^\infty (n\ln n)^{-1}$ diverge?
10777
Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
28254
Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$?
24108
Proof of the equality $\sum\limits_{k=1}^{\infty} \frac{k^2}{2^k} = 6$
82562
Prove that $\sum_{n=2}^\infty (\ln n)^{- \ln n}$ converges
8705
Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
2678
Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
5272
Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$