query: Prove the convergence of the sequence $a_{1} = 4$, $a_{n + 1} = \frac{a_{n}}{2} + \frac{2}{a_{n}}$, $n = 1, 2, \ldots$
positive_passages.docid: 79266
positive_passages.text: Proving a sequence defined by a recurrence relation converges
negative_passages.docid: 116414
negative_passages.text: Proving that the difference of the terms of a sequence, multiplied by $n$ converge
negative_passages.docid: 40797
negative_passages.text: $(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$
negative_passages.docid: 17231
negative_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: 155695
negative_passages.text: The convergence of a sequence
negative_passages.docid: 122942
negative_passages.text: Prove that if $\sum |a_{n}|$ converges then $\sum a_{n}^{2}$ converges
negative_passages.docid: 53001
negative_passages.text: If $|f'(c)| \leq M$ and $ M < 1$, the sequence defined by $a_{n+1} = f(a_n)$ converges
negative_passages.docid: 48500
negative_passages.text: Proof of convergence of sequences
negative_passages.docid: 132358
negative_passages.text: Is the sequence $a_{n+1}=a_n-\frac{1}{a_n}$, $a_0=2$ bounded?
negative_passages.docid: 20909
negative_passages.text: For each positive integer $n$, let $x_n=1/(n+1)+1/(n+2)+\cdots+1/(2n)$. Prove that the sequence $(x_n)$ converges.
negative_passages.docid: 107311
negative_passages.text: Convergence of a sequence and some of its subsequences
negative_passages.docid: 125340
negative_passages.text: Prove the convergence of a series.
negative_passages.docid: 35885
negative_passages.text: Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$
negative_passages.docid: 149806
negative_passages.text: Is it possible to utilize the convergence of the sequence $z_{n+1}=a/(1+z_n)$ to prove that the sequence $x_{n+2} = \sqrt{x_{n+1} x_n}$ is convergent?
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: 50087
negative_passages.text: Convergence of $a_{n+1}=\frac{1}{1+a_n}$
negative_passages.docid: 55464
negative_passages.text: Convergence of sequence given by $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$