query: Evaluate $x_n=\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}.$
positive_passages.docid: 13617
positive_passages.text: Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $.
negative_passages.docid: 123295
negative_passages.text: How does one evaluate $\sqrt[3]{x + iy} + \sqrt[3]{x - iy}$?
negative_passages.docid: 9541
negative_passages.text: Evaluate $\lim_{n\to\infty}\frac {qn+1}{qn} \frac {qn+p+1}{qn+p} \cdots \frac {qn+np+1}{qn+np}$ where $p\in\mathbb{N}\setminus\{0,1\},q>0.$
negative_passages.docid: 154185
negative_passages.text: Evaluate $\iiint_{[0,1]^3}\frac{dx\,dy\,dz}{(1+x^2+y^2+z^2)^2}$
negative_passages.docid: 21457
negative_passages.text: Prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction.
negative_passages.docid: 72502
negative_passages.text: Evaluate $\sum_{n=1}^\infty nx^{n-1}$
negative_passages.docid: 23726
negative_passages.text: A closed form for $1^{2}-2^{2}+3^{2}-4^{2}+ \cdots + (-1)^{n-1}n^{2}$
negative_passages.docid: 25984
negative_passages.text: Explicit solution of the recursion $x_n = x_{n-1}^2 - 2$ with $x_0>2$
negative_passages.docid: 120954
negative_passages.text: Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
negative_passages.docid: 56902
negative_passages.text: Evaluate $\lim_{n \to \infty} \frac{1}{n}[(n +1)(n+2) \cdots (n+n)]^{\frac{1}{n}}$
negative_passages.docid: 74898
negative_passages.text: $1^n +2^n + \cdots +(p-1)^n \mod p =$?
negative_passages.docid: 8589
negative_passages.text: $\sqrt[b_1]{a_1}+\sqrt[b_2]{a_2}+\cdots+\sqrt[b_k]{a_k}$ not an integer
negative_passages.docid: 110955
negative_passages.text: Find the coefficient of $x^9$ in $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})$
negative_passages.docid: 100375
negative_passages.text: Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$.
negative_passages.docid: 14951
negative_passages.text: Other Idea to show an inequality $\dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt n}\geq \sqrt n$
negative_passages.docid: 117320
negative_passages.text: Why is this true: $\|x\|_1 \le \sqrt n \cdot \|x\|_2$?
negative_passages.docid: 87589
negative_passages.text: Evaluate $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+...+\binom{n}{2k}$