How can we show that $\frac{x}{1-x} = \sum \frac{x^{2^n}}{1-x^{2^{n+1}}}$ for $0<x<1$?
44928
Show $\sum_n \frac{z^{2^n}}{1-z^{2^{n+1}}} = \frac{z}{1-z}$
142132
How do you prove $\sum \frac {n}{2^n} = 2$?
24469
Prove $ 2<(1+\frac{1}{n})^{n}$
93427
Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
725
proof of $\sum_{n=1}^\infty n \cdot x^n= \frac{x}{(x-1)^2}$
128285
Why is $ \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$?
37871
Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
2877
Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$
91784
Calculating $\sum_{n=1}^\infty {\frac{1}{2^nn(3n-1)}}$
14239
How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
37007
Evaluating $\sum_{k=1}^{n} \frac{1}{k(k+1)}$
14223
Find the infinite sum $\sum_{n=1}^{\infty}\frac{1}{2^n-1}$
14968
How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$
133505
Let $f(n)=\sum_{k=0}^{\left\lfloor n/2\right\rfloor} {2k \choose k}{n \choose 2k}$ . Show that $\sum_{n\geq 0}^{} f(n)x^n=\frac{1}{\sqrt{1-2x-3x^2}}$
157898
Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$
131023
Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$
36083
If $k > 1$, then $\frac1{(k - 1)^2}-\frac1{(k + 1)^2}=\frac{4k}{(k^2 - 1)^2}$, hence simplify $\sum\limits_{k = 2}^n\frac k{(k^2 - 1)^2}$