query: Prove $\sum\limits_{r=0}^n{n\choose r}^2={2n\choose n}$
positive_passages.docid: 143113
positive_passages.text: Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
negative_passages.docid: 130543
negative_passages.text: If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
negative_passages.docid: 153612
negative_passages.text: Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$
negative_passages.docid: 92046
negative_passages.text: Proving $\sum\limits_{i=0}^n  i 2^{i-1} = (n+1) 2^n - 1$ by induction
negative_passages.docid: 111098
negative_passages.text: Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
negative_passages.docid: 71797
negative_passages.text: Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
negative_passages.docid: 49928
negative_passages.text: Proving limit of a sum
negative_passages.docid: 106281
negative_passages.text: Prove $\lim\limits_{n\to \infty}\frac{1}{\sqrt n}\left|\sum\limits_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}$
negative_passages.docid: 14175
negative_passages.text: $\sum_\limits{n=0}^{\infty} a_n$ converges $\implies \sum_\limits{n=0}^{\infty} a_n^2$ converges
negative_passages.docid: 15045
negative_passages.text: Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$
negative_passages.docid: 10777
negative_passages.text: Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
negative_passages.docid: 36786
negative_passages.text: Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
negative_passages.docid: 141446
negative_passages.text: Proving by induction that $\sum\limits_{k=2}^n \frac1{k^2}\le 1$
negative_passages.docid: 61963
negative_passages.text: Proving that $\sum\limits_{n=0}^{\infty }\frac{1}{(2n)!!}=\sqrt{e}$
negative_passages.docid: 67084
negative_passages.text: Prove or disprove that $\lim_{n \to \infty} {n \choose k} / a^n = 0$ for all $k \in \mathbb{Z_+}, a \in \mathbb{R}: a> 1$.
negative_passages.docid: 17638
negative_passages.text: Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
negative_passages.docid: 1125
negative_passages.text: Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$