How Prove this $ \sum\limits_{k=1}^{2n} { 2n \choose k } { 2n \choose 2n-k } = 2^{4n} - 1$
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
156564
Prove the identity $\sum_{{{\underset{k-even}{k=0}}}}^{n}{n \choose k}2^{k}=\frac{3^{n}+(-1)^{n}}{2}$
111098
Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
92784
Intuitive (combinatorial) proof of $2^n=\sum_{k=0}^n {n\choose k}$
17157
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
5650
Show $\ln2 = \sum_\limits{n=1}^\infty\frac1{n2^n}$
60031
Showing $\sum\limits_{j=0}^M \frac{M \choose j}{N+M \choose j} = \frac{N+M+1}{N+1}$
66568
Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$
107397
Algebraic proof for the identity $\sum_{k=0}^m(-1)^{m-k}{{n \choose {k}}}= {{n-1}\choose m}$
44328
Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
1125
Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
117873
Combinatorially showing $\lim_{n\to \infty}{\frac{2n\choose n}{4^n}}=0$
54595
Prove $\sum\limits_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction.
10211
Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$
130543
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
136580
Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$