query: Prove that: $\sum\limits_{n=1}^\infty \frac{n^2}{2^n}=6$.
positive_passages.docid: 24108
positive_passages.text: Proof of the equality $\sum\limits_{k=1}^{\infty} \frac{k^2}{2^k} = 6$
negative_passages.docid: 17056
negative_passages.text: How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n} $
negative_passages.docid: 24373
negative_passages.text: Hint on computing the series $\sum_{n=2}^\infty \frac{1}{n^2-1}$.
negative_passages.docid: 63968
negative_passages.text: How to prove that $\lim\limits_{n\rightarrow \infty} \frac{1}{n^2}\sum\limits_{k=1}^{n}(n \bmod k)=1-\frac{\pi^2}{12}$?
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: 66568
negative_passages.text: Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$
negative_passages.docid: 62443
negative_passages.text: Prove that $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$ converges and find its sum
negative_passages.docid: 49928
negative_passages.text: Proving limit of a sum
negative_passages.docid: 131023
negative_passages.text: Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$
negative_passages.docid: 20271
negative_passages.text: Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
negative_passages.docid: 117843
negative_passages.text: Finding the sum of series $\sum_{n=0}^∞ \frac{2^n + 3^n}{6^n}$
negative_passages.docid: 126304
negative_passages.text: Determining bounds for the sum $\sum\limits_{n=1}^\infty \frac{1}{2^n - 3^n }$
negative_passages.docid: 9435
negative_passages.text: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.
negative_passages.docid: 17332
negative_passages.text: How to compute $\sum\limits_{n=1}^{\infty}\frac{1}{x^n-1}$?
negative_passages.docid: 78712
negative_passages.text: Prove that $\frac{1}{\sin^2 z } = \sum\limits_{n= -\infty} ^ {+\infty} \frac{1}{(z-\pi n)^2} $
negative_passages.docid: 151477
negative_passages.text: computing the series $\sum_{n=1}^\infty \frac{1}{n^2 2^n}$
negative_passages.docid: 137640
negative_passages.text: Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$