query: What is the closed form of $\sum_{i=1}^n\frac{2i}{2^i}$
positive_passages.docid: 562
positive_passages.text: Formula for calculating $\sum_{n=0}^{m}nr^n$
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: 48200
negative_passages.text: Simplify $\sum_{i=0}^n (i+1)\binom ni$
negative_passages.docid: 5537
negative_passages.text: Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
negative_passages.docid: 29524
negative_passages.text: Formula for the sum $\sum_{i=2}^{n} \frac1{i^2-1}$
negative_passages.docid: 151085
negative_passages.text: Simplify the sum $ \sum_{i=0}^{k}(-1)^i i \binom{n}{i} \binom{n}{k-i}$
negative_passages.docid: 153226
negative_passages.text: Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
negative_passages.docid: 44928
negative_passages.text: Show $\sum_n \frac{z^{2^n}}{1-z^{2^{n+1}}} = \frac{z}{1-z}$
negative_passages.docid: 116491
negative_passages.text: Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$
negative_passages.docid: 45862
negative_passages.text: Nested sum of binomial coefficient in closed form
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: 70841
negative_passages.text: A simple series $\sum_{i=1}^\infty \frac{i}{2^i} = 2$
negative_passages.docid: 36742
negative_passages.text: The sum of $(-1)^n \frac{\ln n}{n}$
negative_passages.docid: 146762
negative_passages.text: Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$
negative_passages.docid: 107976
negative_passages.text: Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$
negative_passages.docid: 98808
negative_passages.text: Finite Sum $\sum_{i=1}^n\frac i {2^i}$
negative_passages.docid: 151477
negative_passages.text: computing the series $\sum_{n=1}^\infty \frac{1}{n^2 2^n}$