query: Show $a_{n+1}=\sqrt{2+a_n},a_1=\sqrt2$ is monotone increasing and bounded by $2$; what is the limit?
positive_passages.docid: 686
positive_passages.text: $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$
negative_passages.docid: 31174
negative_passages.text: Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$
negative_passages.docid: 121544
negative_passages.text: Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
negative_passages.docid: 97047
negative_passages.text: Limit for sequence $a_{m+n}\leq a_m+a_n$
negative_passages.docid: 26783
negative_passages.text: If $(1+\sqrt{2})^n=a_{n}+b_{n}\sqrt{2}\;,(\forall n\in \mathbb{N}).$Then $\lim_{n\rightarrow \infty}\frac{a_{n}}{b_{n}} = $
negative_passages.docid: 116901
negative_passages.text: Monotonicity of $\frac{n}{\sqrt[n]{(n!)}}$
negative_passages.docid: 114506
negative_passages.text: Expansion of $(1+\sqrt{2})^n$
negative_passages.docid: 122413
negative_passages.text: Show $b_n=\sqrt[k]{n+1}-\sqrt[k]{n}$ converges towards $0$ for all $k \geq 2$.
negative_passages.docid: 64329
negative_passages.text: Limit of the recursive sequence $a_{n+1} = \frac{1}{2}(a_n + \frac{p}{a_n})$
negative_passages.docid: 100048
negative_passages.text: Got stuck in proving monotonic increasing of a recurrence sequence
negative_passages.docid: 150995
negative_passages.text: Show that $\liminf(\sqrt{x_n})=\sqrt{\liminf x_n}$
negative_passages.docid: 17289
negative_passages.text: prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit
negative_passages.docid: 26837
negative_passages.text: Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?
negative_passages.docid: 97656
negative_passages.text: Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$
negative_passages.docid: 67638
negative_passages.text: Show that $x_{n+1}=x_n(2-ax_n)$ converges and find the limit
negative_passages.docid: 118244
negative_passages.text: An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$
negative_passages.docid: 17090
negative_passages.text: Compute $\lim_\limits{n\to\infty}a_n$ where $a_{n+2}=\sqrt{a_n.a_{n+1}}$