query: Sequence $x_{n}=\frac{1}{2}(x_{n-1}+\frac{a}{x_{n-1}}).$
positive_passages.docid: 13390
positive_passages.text: Recursive Monotone Decreasing Sequence Proof ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$
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: 14836
negative_passages.text: A polynomial sequence
negative_passages.docid: 8814
negative_passages.text: Proving that a sequence such that $|a_{n+1} - a_n| \le 2^{-n}$ is Cauchy
negative_passages.docid: 73176
negative_passages.text: $\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in...
negative_passages.docid: 22077
negative_passages.text: Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$
negative_passages.docid: 76013
negative_passages.text: Evaluating a sequence of complex numbers: $|z_n - z_m| < \frac1{1+ |n-m|}$
negative_passages.docid: 45885
negative_passages.text: If there exist sequence such that $g(x_n)=f(x_{n+1})$, then we have $g(x_0)=f(x_0)$ for some $x_0$
negative_passages.docid: 34025
negative_passages.text: Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$
negative_passages.docid: 143424
negative_passages.text: Sequence problem involving inequalities
negative_passages.docid: 71639
negative_passages.text: Proof $(1+1/n)^n$ is an increasing sequence
negative_passages.docid: 141463
negative_passages.text: Convergence of $x_{n+1} = \frac12(x_n + \frac2{x_n}).$
negative_passages.docid: 70746
negative_passages.text: Proving that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.
negative_passages.docid: 105882
negative_passages.text: Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$
negative_passages.docid: 2471
negative_passages.text: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
negative_passages.docid: 62937
negative_passages.text: Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$
negative_passages.docid: 100232
negative_passages.text: Find equivalent of recurrent sequence $u_1=1$, $u_{n+1}=1+\frac n {u_n}$