Sum of the series $~\sum_{k=0}^{n}k \cdot {2n \choose k} .$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
125148
Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$
12812
Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $
114911
Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$
155611
Sum of series $\frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{4 \cdot 5 \cdot 6} + \frac{1}{6 \cdot 7 \cdot 8} + \cdots $
92784
Intuitive (combinatorial) proof of $2^n=\sum_{k=0}^n {n\choose k}$
109599
A finite sum involving the binomial coefficients and the harmonic numbers
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
148883
Sum of the series $\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\dots$
127806
How to find the sum of the series $\sum_{k=2}^\infty \frac{1}{k^2-1}$?
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-
125933
On the series $\sum \limits_{n=2}^{\infty} \frac{(-1)^n}{n \log n}$
32791
Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$
96945
General formula of $\sum_{n\ge1}{\binom{2n}{n}^{-k}}$.
81589
Find the sum of $\sum_{k=0}^{\infty} 1/[(r+2k+2)(s+2k+2)]$
136580
Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$