query: Does $\frac12+\frac14+\frac18+\dots$ equal $1$?
positive_passages.docid: 147165
positive_passages.text: Does the fact that $\sum_{n=1}^\infty 1/2^n$ converges to $1$ mean that it equals $1$?
negative_passages.docid: 49851
negative_passages.text: when the sum of some fractions be $1$
negative_passages.docid: 127537
negative_passages.text: Is there any mathematical or physical situations that $1+2+3+\ldots\infty=-\frac{1}{12}$ shows itself?
negative_passages.docid: 64941
negative_passages.text: Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?
negative_passages.docid: 158210
negative_passages.text: showing $\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$
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: 11664
negative_passages.text: If $a_0+\frac{a_1}{2}+\dots + \frac{a_n}{n+1}=0$ then $a_nx^n+\dots + a_0$ has a root in $[0,1]$
negative_passages.docid: 148461
negative_passages.text: If $S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}$, then calculate $14S$.
negative_passages.docid: 4278
negative_passages.text: Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?
negative_passages.docid: 77978
negative_passages.text: Show that $\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x)$
negative_passages.docid: 106028
negative_passages.text: Sum of $\frac{1}{(2n)(2n+1)}$ from $1$ to infinity
negative_passages.docid: 5358
negative_passages.text: How to prove that 2017 divides $1^{2017}+2^{2017}+\dots+2017^{2017}$?
negative_passages.docid: 6717
negative_passages.text: Evaluation of $\int_{\frac{1}{2014}}^{2014} \frac{\tan^{-1}x}{x} dx$
negative_passages.docid: 50599
negative_passages.text: Prove that $\frac{1}{1*3}+\frac{1}{3*5}+\frac{1}{5*7}+...+\frac{1}{(2n-1)(2n+1)}=\frac{n}{2n+1}$
negative_passages.docid: 23576
negative_passages.text: Calculating $1+\frac13+\frac{1\cdot3}{3\cdot6}+\frac{1\cdot3\cdot5}{3\cdot6\cdot9}+\frac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $
negative_passages.docid: 140342
negative_passages.text: $\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$
negative_passages.docid: 136994
negative_passages.text: Proving :$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$