query: $a_{n+1}=\frac2{a_n+a_{n−1}}$ . Prove that this sequence has a limit.
positive_passages.docid: 145863
positive_passages.text: sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
negative_passages.docid: 112430
negative_passages.text: Prove that sequence $ x_{(n+1)}= \frac {a}{1+x_n}$ is convergent to positive root of $x^2+x-a=0,$where $a >0$ and $x_1 >0$
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: 13486
negative_passages.text: Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$
negative_passages.docid: 25206
negative_passages.text: How to prove that $\lim\limits_{n\to \infty}a_{n+1}=\lim\limits_{n\to \infty}a_{n}$
negative_passages.docid: 27410
negative_passages.text: If the sequence satisfies the property lim$_{n\to \infty}(a_n-a_{n-2})=0$, prove that lim$_{n\to \infty}\frac{a_n-a_{n-1}}{n}=0$.
negative_passages.docid: 27932
negative_passages.text: Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$
negative_passages.docid: 148782
negative_passages.text: How to prove these two limits $\lim_{n\to\infty}\frac{a_{n}}{n}$ and $\lim_{n\to\infty}\frac{a_{n+1}-a_{n}}{n}$ exist?
negative_passages.docid: 12031
negative_passages.text: Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
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: 135877
negative_passages.text: $\lim\limits_{n \to \infty} \frac{a_n}{b_n}=l$ ; Prove $\sum\limits_{k=1}^\infty a_k$ converges iff $\sum\limits_{k=1}^\infty b_k$ converges.
negative_passages.docid: 118023
negative_passages.text: Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of prime numbers, then $\lim_{n \to \infty}\frac{p_{n+1}}{p_n} = 1$?
negative_passages.docid: 49497
negative_passages.text: $\lim a_n = 0$ if $\left|\frac{a_{n +1}}{a_n}\right|\to L < 1$
negative_passages.docid: 121606
negative_passages.text: How do I prove that the limit of the sequence exists: $a_n = \frac1n +\frac 1{n+1} + \frac1{n+2} + \dots + \frac1{2n}$?
negative_passages.docid: 17779
negative_passages.text: Proving the Limit of a Sequence of without the Dominated Convergence Theorem
negative_passages.docid: 4345
negative_passages.text: Prove that lim $(\sqrt{n^2+n}-n) = \frac{1}{2}$
negative_passages.docid: 95095
negative_passages.text: $x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.