query: Find the sum of first $n$ terms of the series: $\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+\cdots +\frac{1}{n\times(n+1)}$
positive_passages.docid: 14306
positive_passages.text: What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$
negative_passages.docid: 92557
negative_passages.text: If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \frac{n-1}{n+1}$, find $a_n$ and the sum.
negative_passages.docid: 94943
negative_passages.text: Sum of series with addition: $\sum_{n=1}^\infty \frac{1}{n^2(n+1)}$
negative_passages.docid: 42271
negative_passages.text: Infinite Series $1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$
negative_passages.docid: 58262
negative_passages.text: Find the sum of the series $1+\frac{1}{3}\cdot\frac{1}{4}+\frac{1}{5}\cdot\frac{1}{4^2}+\frac{1}{7}\cdot\frac{1}{4^3}+\cdots$
negative_passages.docid: 28378
negative_passages.text: Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$
negative_passages.docid: 55339
negative_passages.text: Find the sum $\sum_{n=1}^{\infty}\frac{ 1}{n^{2}(2n-1)}$
negative_passages.docid: 93690
negative_passages.text: How to obtain the sum of the following series? $\sum_{n=1}^\infty{\frac{n^2}{2^n}}$
negative_passages.docid: 133420
negative_passages.text: Computing the binomial series: $\binom{n}{2}\cdot2^2+\binom{n}{4}\cdot2^4+\binom{n}{6}\cdot2^6+\cdots+ \binom{n}{n}\cdot2^n$
negative_passages.docid: 43903
negative_passages.text: Find the sum for $\sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{2n+1}{3^n} $
negative_passages.docid: 74120
negative_passages.text: Find the value of $\sum_{n=1}^{\infty} \frac{2}{n}-\frac{4}{2n+1}$
negative_passages.docid: 100936
negative_passages.text: What is the sum of this series: $\frac{2}{\pi}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$
negative_passages.docid: 30518
negative_passages.text: Formula for $\sum_{k=1}^n \frac{1}{k(k+1)(k+2)}$?
negative_passages.docid: 62271
negative_passages.text: Find the sum of the series: $\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+....+\cos^3 {(2n-1)\alpha}$.
negative_passages.docid: 29170
negative_passages.text: Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$
negative_passages.docid: 47706
negative_passages.text: Find the value of $ \sum_{n=1}^{\infty} \frac{n^3}{3^n} $
negative_passages.docid: 51146
negative_passages.text: Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$