query: Compute closed form of $\sum_{k = 1}^{\infty} \frac{k}{3^{k}}$
positive_passages.docid: 6416
positive_passages.text: How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
negative_passages.docid: 102942
negative_passages.text: Compute the sum $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$
negative_passages.docid: 23726
negative_passages.text: A closed form for $1^{2}-2^{2}+3^{2}-4^{2}+ \cdots + (-1)^{n-1}n^{2}$
negative_passages.docid: 42736
negative_passages.text: Is there a closed form for $\sum_{k=0}^n \frac{x^k}{k!}$?
negative_passages.docid: 99314
negative_passages.text: Closed form of infinite product $ \prod\limits_{k=0}^\infty \left(1+\frac{1}{2^{2^k}}\right)$
negative_passages.docid: 17686
negative_passages.text: Evaluate the infinite series $\sum_{r=1}^{\infty} \frac{2^{-r}}{r^{2}}$
negative_passages.docid: 78473
negative_passages.text: Showing $ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}}$
negative_passages.docid: 39909
negative_passages.text: Getting a closed form from $\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1$
negative_passages.docid: 60104
negative_passages.text: Evaluate $\sum_{k=1}^\infty\frac{x^k}{k(k+1)}$
negative_passages.docid: 129734
negative_passages.text: Computing $\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$
negative_passages.docid: 125148
negative_passages.text: Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$
negative_passages.docid: 99843
negative_passages.text: How to simplify $ \sum_{k=0}^{\infty} \binom{k + r - 1}{k} x^k$
negative_passages.docid: 49052
negative_passages.text: Find $\sum_{n=1}^\infty \frac{n^3}{n!}$.
negative_passages.docid: 32866
negative_passages.text: How find this sum $\sum\limits_{n=0}^{\infty}\frac{1}{(3n+1)(3n+2)(3n+3)}$
negative_passages.docid: 9610
negative_passages.text: Find the sum of $\sum\limits_{n=1}^{\infty} \frac{x^{n}}{n(n+1)}$
negative_passages.docid: 69084
negative_passages.text: What is the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}$?
negative_passages.docid: 24373
negative_passages.text: Hint on computing the series $\sum_{n=2}^\infty \frac{1}{n^2-1}$.