query: Evaluating $ \sum_{i=0}^{\infty}ir^i$ for $|r|\lt 1$
positive_passages.docid: 6416
positive_passages.text: How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
negative_passages.docid: 10675
negative_passages.text: Proving that $\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^2dt=\sum_{n=0}^{\infty}|a_n|^2r^{2n}$
negative_passages.docid: 154286
negative_passages.text: Evaluating $\lim_{n \to \infty} (1 + 1/n)^{n}$
negative_passages.docid: 140458
negative_passages.text: Infinite summation involving exponents
negative_passages.docid: 108010
negative_passages.text: Evaluating $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$
negative_passages.docid: 33965
negative_passages.text: Convergence of $\sum_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$
negative_passages.docid: 146651
negative_passages.text: Evaluting sum $\sum \limits_{n=0}^\infty\frac{n^k}{n!}$
negative_passages.docid: 13569
negative_passages.text: Ways of showing $\sum_\limits{n=1}^{\infty}\ln(1+1/n)$ to be divergent
negative_passages.docid: 913
negative_passages.text: Evaluating $\sum_{n=0}^{\infty}{\sin^{3}\left(3^{n}\right) \over 3^{n}}$
negative_passages.docid: 94143
negative_passages.text: Evaluate:: $ 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n+1}\left( 1 + \frac12 +\cdots + \frac 1n\right) $
negative_passages.docid: 70841
negative_passages.text: A simple series $\sum_{i=1}^\infty \frac{i}{2^i} = 2$
negative_passages.docid: 12480
negative_passages.text: Convergence of $\sum_{n=1}^{\infty} \frac{\frac{1}{2}+(-1)^n}{n}$
negative_passages.docid: 32423
negative_passages.text: Convergence of $\sum _{k=1}^\infty (1-\frac{1}{k})^{k^2}$
negative_passages.docid: 124426
negative_passages.text: Help in evaluating: $\lim_{n \to \infty} \frac{\sum_{k=1}^n\lfloor kx\rfloor}{n^2} $
negative_passages.docid: 38062
negative_passages.text: Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$
negative_passages.docid: 85285
negative_passages.text: Show $\sum_{i=1}^\infty x_i y_i$ is absolutely convergent
negative_passages.docid: 121337
negative_passages.text: Divergence of $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$?