query: Show that $\sum\limits_n1/x_{n}^{2} = 1/10$ where $x_{n}$ is the $n^{\text{th}}$ positive root of $\tan x = x$
positive_passages.docid: 24222
positive_passages.text: Sum of the squares of the reciprocals of the fixed points of the tangent function
negative_passages.docid: 21791
negative_passages.text: Show that $\sum\limits_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\frac{1}{p!p} $ for every positive integer $p$
negative_passages.docid: 30672
negative_passages.text: Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
negative_passages.docid: 80277
negative_passages.text: How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $?
negative_passages.docid: 82085
negative_passages.text: Without solving the equation, show that all nine roots of $(𝑧 − 1)^{10} = 𝑧^{10}$ lie on the line $Re (𝑧) = \frac{1}{2}$
negative_passages.docid: 13569
negative_passages.text: Ways of showing $\sum_\limits{n=1}^{\infty}\ln(1+1/n)$ to be divergent
negative_passages.docid: 85626
negative_passages.text: prove $\sum_{i=1}^{n}(x_i-\bar{x})^2\lt\sum_{i=1}^{n}(x_i-a)^2$
negative_passages.docid: 42251
negative_passages.text: Limit of non-negative real root of $x^n+x^{n-1}+x-1=0$ as $n \to \infty$
negative_passages.docid: 68124
negative_passages.text: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
negative_passages.docid: 11059
negative_passages.text: Verify the identity: $\tan^{-1} x +\tan^{-1} (1/x) = \pi /2$
negative_passages.docid: 36060
negative_passages.text: If $\lim_{n\to\infty}x_n=\alpha$, show $\lim_{n\to\infty}(x_1x_2\dots x_n)^{1/n} =\alpha$.
negative_passages.docid: 142360
negative_passages.text: How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
negative_passages.docid: 65249
negative_passages.text: An interesting exercise about converging positive series, involving $\sum_{n\geq 1}a_n^{\frac{n-1}{n}}$
negative_passages.docid: 65698
negative_passages.text: Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$
negative_passages.docid: 64336
negative_passages.text: Show $\sum_{k=1}^n \frac{1}{k^2} \le 2$ and $\ln(n!) \ge 1 -n+n\ln(n)$ for all positive integers n
negative_passages.docid: 151111
negative_passages.text: How do I show that $\frac{n}{\varphi(n)}=\sum\limits_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}$?
negative_passages.docid: 80440
negative_passages.text: Limit of $(x_n)$ with $0<x_1<1$ and $x_{n + 1} = x_n - x_n^{n + 1}$