query: Show that for each $n \geq 2$, $\left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{9}\right) \cdots \left(1 - \frac{1}{n^2}\right) = \frac{n + 1}{2n}$
positive_passages.docid: 73002
positive_passages.text: For $n \geq 2$, prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
negative_passages.docid: 45742
negative_passages.text: Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
negative_passages.docid: 63553
negative_passages.text: Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$
negative_passages.docid: 26962
negative_passages.text: Proving an inequality involving the logarithm function: $\frac{1}{n+1} \leq \ln \left(1+\frac 1n\right) \leq \frac 1n$
negative_passages.docid: 112929
negative_passages.text: How do I prove $\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$
negative_passages.docid: 44940
negative_passages.text: Proving by induction that $1^3 + 2^3 + 3^3 + \ldots + n^3 = \left[\frac{n(n+1)}{2}\right]^2$
negative_passages.docid: 104807
negative_passages.text: Prove the inequality $\left|\frac{m}{n}-\frac{1+\sqrt{5}}{2}\right|<\frac{1}{mn}$
negative_passages.docid: 25054
negative_passages.text: Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ by Induction
negative_passages.docid: 33903
negative_passages.text: Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
negative_passages.docid: 96039
negative_passages.text: Why is $\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}\le \frac 1 {n+1}$
negative_passages.docid: 34349
negative_passages.text: Prove that $\frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n} \geq \frac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$
negative_passages.docid: 123523
negative_passages.text: Showing $\left(1+\frac{1}{n}\right)^n$ is an increasing sequence by comparing the binomial expansions term by term
negative_passages.docid: 146546
negative_passages.text: Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$
negative_passages.docid: 81794
negative_passages.text: Prove that $(ab,cd)=(a,c)(b,d)\left(\frac{a}{(a,c)},\frac{d}{(b,d)}\right)\left(\frac{c}{(a,c)},\frac{b}{(b,d)}\right)$
negative_passages.docid: 82591
negative_passages.text: Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
negative_passages.docid: 80826
negative_passages.text: To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$