query: Compute $1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \cdots + n \cdot \frac {1}{2^n} + \cdots $
positive_passages.docid: 6416
positive_passages.text: How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
negative_passages.docid: 55061
negative_passages.text: Evaluating the sum after reordering an infinite series $1-\frac {1} {2}+\frac {1} {3}-\frac {1} {4}+\ldots $
negative_passages.docid: 91410
negative_passages.text: Finding the sum of this series: $1+\frac 12 + \frac 13 + \cdots + \frac 1{50}$
negative_passages.docid: 18796
negative_passages.text: How to find the sum of the series $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$?
negative_passages.docid: 155611
negative_passages.text: Sum of series $\frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{4 \cdot 5 \cdot 6} + \frac{1}{6 \cdot 7 \cdot 8} + \cdots $
negative_passages.docid: 112036
negative_passages.text: sum of series $\frac{n}{1\cdot 2 \cdot 3}+\frac{n-1}{2\cdot 3\cdot 4}+\frac{n-2}{3\cdot 4 \cdot 5}+\cdots \cdots n$ terms
negative_passages.docid: 98933
negative_passages.text: Calculating the sum of $f(x) = \sum_{n=0}^{\infty} {n \cdot 2^n \cdot x^n}$
negative_passages.docid: 73191
negative_passages.text: Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$
negative_passages.docid: 7063
negative_passages.text: How to compute the  formula $\sum  \limits_{r=1}^d r \cdot 2^r$?
negative_passages.docid: 18814
negative_passages.text: Sum of the series $1+\frac{1\cdot 3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+\cdots$
negative_passages.docid: 38705
negative_passages.text: Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$
negative_passages.docid: 100896
negative_passages.text: What is the sum of this? $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$
negative_passages.docid: 114212
negative_passages.text: How to use Mathematical Induction to prove $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{n(n + 1)} = \frac{n}{n + 1}$?
negative_passages.docid: 145444
negative_passages.text: Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$
negative_passages.docid: 112374
negative_passages.text: Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$
negative_passages.docid: 84335
negative_passages.text: Solving for $\sum_{n = 1}^{\infty} \frac{n^3}{8^n}$?
negative_passages.docid: 120430
negative_passages.text: A question about how to express a fraction as ${1\over q_1}+{1\over q_2}+ \cdots+{1\over q_N}$