query: Why is $\sqrt{2\sqrt{2\sqrt{2\cdots}}} = 2$?
positive_passages.docid: 17231
positive_passages.text: How I can prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2?
negative_passages.docid: 142470
negative_passages.text: $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ??
negative_passages.docid: 110661
negative_passages.text: When does $\sqrt{a b} = \sqrt{a} \sqrt{b}$?
negative_passages.docid: 87411
negative_passages.text: Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$
negative_passages.docid: 23048
negative_passages.text: The ring $\mathbb Z[\sqrt{-2}]= \{a+b\sqrt{-2} ; a\in \mathbb Z,b\in \mathbb Z \}$ has a Euclidean algorithm
negative_passages.docid: 66082
negative_passages.text: Using $\sqrt{1-t}\leq 1-\frac t2$ to show that $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}\geq\frac1{2\sqrt n}$
negative_passages.docid: 118981
negative_passages.text: Proving $\sqrt{ab} = \sqrt a\sqrt b$
negative_passages.docid: 6582
negative_passages.text: Prove the existence of the square root of $2$.
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: 1401
negative_passages.text: $\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation
negative_passages.docid: 27922
negative_passages.text: prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
negative_passages.docid: 73099
negative_passages.text: Why is $d(x^2)= 2xdx$?
negative_passages.docid: 127830
negative_passages.text: proof that $\sqrt(2)$ is irrational / why is the fraction irreducible
negative_passages.docid: 75679
negative_passages.text: How to prove that $2 \arctan\sqrt{x} = \arcsin \frac{x-1}{x+1} + \frac{\pi}{2}$
negative_passages.docid: 83753
negative_passages.text: Why $\,p \cdot (p^{-1} \mod q) + q \cdot (q^{-1} \mod p) = pq + 1$?
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.
negative_passages.docid: 107914
negative_passages.text: Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$