query: How do you prove $\sum \frac {n}{2^n} = 2$?
positive_passages.docid: 28254
positive_passages.text: Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$?
negative_passages.docid: 68400
negative_passages.text: Proof that $\frac{(2n)!}{2^n}$ is integer
negative_passages.docid: 108318
negative_passages.text: counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
negative_passages.docid: 143337
negative_passages.text: How to prove that $\sum_{i=0}^n 2^i\binom{2n-i}{n} = 4^n$.
negative_passages.docid: 7686
negative_passages.text: How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$?
negative_passages.docid: 150642
negative_passages.text: How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?
negative_passages.docid: 139229
negative_passages.text: How to prove by arithmetical means that $\sum\limits_{k=1}^\infty \frac{((k-1)!)^2}{(2k)!} =\frac{1}{3}\sum\limits_{k=1}^{\infty}\frac{1}{k^{2}}$
negative_passages.docid: 111098
negative_passages.text: Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
negative_passages.docid: 151477
negative_passages.text: computing the series $\sum_{n=1}^\infty \frac{1}{n^2 2^n}$
negative_passages.docid: 117056
negative_passages.text: How to show $\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}=2$?
negative_passages.docid: 125701
negative_passages.text: Prove that $-2\log(2) = -2 + \sum_{n=1}^{\infty}\frac{1}{n(2n+1)}$
negative_passages.docid: 78542
negative_passages.text: What is $\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$?
negative_passages.docid: 7201
negative_passages.text: Prove that $x_1^2+x_2^2+x_3^2=1$ yields $ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $
negative_passages.docid: 118372
negative_passages.text: Combination proof for $n(n+1)2^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}$
negative_passages.docid: 139842
negative_passages.text: Seeking a combinatorial proof $\sum\limits _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$
negative_passages.docid: 10917
negative_passages.text: Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
negative_passages.docid: 150148
negative_passages.text: Sum of $\sum^\infty_{n=2} \frac{1}{\ln{(n!)}}$