query: Let $a>0$ and $x_1 > 0$ and $x_{n+1} = \sqrt{a + x_n}$ for $n \in \mathbb{N}$. Show that $\{x_n\}_{n\ge 1}$ converges
positive_passages.docid: 144786
positive_passages.text: Show that the following inductively defined sequence converges and find its limit.
negative_passages.docid: 63562
negative_passages.text: Show that $x_{n+1}-\dfrac{x_n}{2}$ converges to zero implies that $(x_n)_n$ also converges to zero.
negative_passages.docid: 142509
negative_passages.text: If $\alpha\in (0,1]$ and $x_n=\alpha x_{n-1}+(1-\alpha)x_{n-2}$, show that the sequence $\{x_n\}$ is convergent.
negative_passages.docid: 9134
negative_passages.text: Suppose $\sum{a_n}$ converges. How do I prove that $\sum{\frac{\sqrt{a_n}}{n}}$ converges
negative_passages.docid: 142057
negative_passages.text: Prove that if $\sum_{n=1}^\infty a_n$ converges then $\sum_{k=1}^\infty A_k$ converges
negative_passages.docid: 42292
negative_passages.text: Show that $a_n<\sqrt{2}$ for every $n\in\mathbb{N}$.
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: 837
negative_passages.text: Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.
negative_passages.docid: 59475
negative_passages.text: Show that $\sqrt{n+1}-\sqrt{n}\to0$
negative_passages.docid: 87834
negative_passages.text: How can we show that the following sequence converges?
negative_passages.docid: 16970
negative_passages.text: Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
negative_passages.docid: 100580
negative_passages.text: Proof that a sequence converges.
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: 55464
negative_passages.text: Convergence of sequence given by $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$
negative_passages.docid: 107399
negative_passages.text: A sequence and convergence
negative_passages.docid: 19225
negative_passages.text: Let $\{a_n\}$ be defined as follows: $a_1 = 2$, $a_{n+1}=\frac{1}{3-a_n}$, if $n \geq 1$. Does $\{a_n\}$ converge?
negative_passages.docid: 40797
negative_passages.text: $(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$