query: nth root of $S_n=\frac{n^n}{(n+1)(n+2)...(n+n)}$
positive_passages.docid: 14013
positive_passages.text: How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)... 2n} = \frac{4}{e}$
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: 111353
negative_passages.text: Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges
negative_passages.docid: 13789
negative_passages.text: solve $T(n)=T(n-1)+T(\frac{n}{2})+n$
negative_passages.docid: 28332
negative_passages.text: Proof of $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$
negative_passages.docid: 128647
negative_passages.text: Calculating square roots using the recurrence $x_{n+1} = \frac12 \left(x_n + \frac2{x_n}\right)$
negative_passages.docid: 4345
negative_passages.text: Prove that lim $(\sqrt{n^2+n}-n) = \frac{1}{2}$
negative_passages.docid: 15164
negative_passages.text: Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
negative_passages.docid: 109304
negative_passages.text: The roots of the equation $z^n=(1+z)^n$...
negative_passages.docid: 133451
negative_passages.text: To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$
negative_passages.docid: 82636
negative_passages.text: Maybe is right $\frac{n^2 + 1}{4k + 3} \notin \mathbb{Z}, n, k \in \mathbb{N}^{+}$
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
negative_passages.docid: 142008
negative_passages.text: Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational
negative_passages.docid: 120093
negative_passages.text: Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction
negative_passages.docid: 43188
negative_passages.text: The $n^{th}$ root of the geometric mean of binomial coefficients.
negative_passages.docid: 109760
negative_passages.text: Showing that the product of all $n$th Roots of Unity is $(-1)^{n+1}$