query: $\frac{1}{1-x} = (1+x)(1+x^2)(1+x^4)(1+x^8)...$
positive_passages.docid: 74204
positive_passages.text: Evaluate $\lim_{n\to\infty}(1+x)(1+x^2)\cdots(1+x^{2n}),|x|<1$
negative_passages.docid: 117816
negative_passages.text: Is $\frac{x^2+x}{x+1}$ a polynomial?
negative_passages.docid: 102019
negative_passages.text: How to solve inequality $1-\frac{1}{x}\geq 0$ correctly?
negative_passages.docid: 56032
negative_passages.text: How come $\frac{1}{1-x}$ and $\frac{x}{1-x}$ have the same derivative?
negative_passages.docid: 136994
negative_passages.text: Proving :$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$
negative_passages.docid: 79170
negative_passages.text: $1 + \frac{1}{1+2}+ \frac{1}{1+2+3}+ ... + \frac{1}{1+2+3+...+n} = ?$
negative_passages.docid: 88670
negative_passages.text: Evaluating $\int \frac{1}{{x^4+1}} dx$
negative_passages.docid: 59175
negative_passages.text: Given that $x+\frac{1}{x}=\sqrt{3}$, find $x^{18}+x^{24}$
negative_passages.docid: 100080
negative_passages.text: Proving with induction $(1-x)^n<\frac 1 {1+nx}$
negative_passages.docid: 36837
negative_passages.text: Prove $\frac{1+a^{2}}{1+b+c^{2}} +\frac{1+b^{2}}{1+c+a^{2}} +\frac{1+c^{2}}{1+a+b^{2}} \geq 2$
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: 24469
negative_passages.text: Prove $ 2<(1+\frac{1}{n})^{n}$
negative_passages.docid: 112171
negative_passages.text: Proposition: $\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = \frac{1 + \sqrt{1 + 4x}}{2}$.
negative_passages.docid: 68990
negative_passages.text: Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$
negative_passages.docid: 149201
negative_passages.text: What does $\lim_{x\to4} \frac{1}{x-4}$ equal?
negative_passages.docid: 114861
negative_passages.text: equation $\frac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x^2}+\frac{1}{y^2}}$
negative_passages.docid: 128309
negative_passages.text: Use Mathematical Induction to prove that $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +...+\frac{1}{n(n+1)}=1-\frac{1}{n+1}$