query: Proving that a sequence converge to $\sqrt{2}$
positive_passages.docid: 128647
positive_passages.text: Calculating square roots using the recurrence $x_{n+1} = \frac12 \left(x_n + \frac2{x_n}\right)$
negative_passages.docid: 112595
negative_passages.text: Prove that the limit of this sequence is $\frac{1}{2}(1 + \sqrt{5})$
negative_passages.docid: 94409
negative_passages.text: showing that the sequence $a_n=1+\frac{1}{2}+...+\frac{1}{n} - \log(n)$ converges
negative_passages.docid: 78886
negative_passages.text: Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$
negative_passages.docid: 145863
negative_passages.text: sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
negative_passages.docid: 29266
negative_passages.text: Limit of a sequence defined by recursive relation : $ a_n = \sqrt{a_{n-1}a_{n-2}}$
negative_passages.docid: 37906
negative_passages.text: Prove $s_n$ is a Cauchy sequence for $|s_{n+1}-s_n|\lt2^{-n}$
negative_passages.docid: 141484
negative_passages.text: Convergence of a sequence by convergence of sub-subsequence
negative_passages.docid: 88365
negative_passages.text: Prove the sequence converge and find its limit
negative_passages.docid: 60024
negative_passages.text: Proof that a sequence is convergent
negative_passages.docid: 129351
negative_passages.text: Show that $\sqrt{n^2+1}-n$ converges to 0
negative_passages.docid: 52711
negative_passages.text: Prove that the sequence $\{v_n\}$ converges to the maximum value of $f$ on $[a,b]$
negative_passages.docid: 114240
negative_passages.text: Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges
negative_passages.docid: 96412
negative_passages.text: Prove $ \sum \frac{\cos n} { \sqrt n}$ converges
negative_passages.docid: 25583
negative_passages.text: Show that the sequence $a_0 = 1$, $a_{n+1 }= \sqrt{2+a_n}$ is monotonically increasing
negative_passages.docid: 50930
negative_passages.text: Proving the continued fraction representation of $\sqrt{2}$
negative_passages.docid: 107953
negative_passages.text: Limit of the sequence $\lim_{n\rightarrow\infty}\sqrt[n]n$