query: Explain $\sum_{n=0}^\infty 2^n= 1+2+4+8+16+\cdots =-1$
positive_passages.docid: 23270
positive_passages.text: The sum of powers of two and two's complement – is there a deeper meaning behind this?
negative_passages.docid: 128200
negative_passages.text: Sequence sum question: $\sum_{n=0}^{\infty}nk^n$
negative_passages.docid: 138492
negative_passages.text: Finding the sum of the infinite series whose general term is not easy to visualize: $\frac16+\frac5{6\cdot12}+\frac{5\cdot8}{6\cdot12\cdot18}+\cdots$
negative_passages.docid: 4253
negative_passages.text: How to calculate: $\sum_{n=1}^{\infty} n a^n$
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: 57759
negative_passages.text: Determining the infinite sum $\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{2n+1}$
negative_passages.docid: 104646
negative_passages.text: Question on simplification of $\sum_{n=1}^{\infty}\frac{2}{(2n+1)(2n+3)}$?
negative_passages.docid: 62913
negative_passages.text: Understanding $\sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}$
negative_passages.docid: 116723
negative_passages.text: Evaluating the sum $\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$
negative_passages.docid: 92810
negative_passages.text: What is $\sum_{r=1}^\infty\frac{r+2}{2^{r+1}(r)(r+1)}$?
negative_passages.docid: 19581
negative_passages.text: $\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$
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: 14487
negative_passages.text: Does $\sum_{n=2}^\infty (n\ln n)^{-1}$ diverge?
negative_passages.docid: 53658
negative_passages.text: Find the sum series $\sum_{n=1}^{+\infty} \frac{(-1)^{n-1}(2n-1)}{2^{n-1}}$
negative_passages.docid: 6416
negative_passages.text: How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
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: 152227
negative_passages.text: How to sum this series to infinity: $\sum_{n=0}^{\infty} \frac1{2^{2^n}}$