query: Convergence issues in the identity $\lim a_n^{1/n}=\lim a_{n+1}/a_n$.
positive_passages.docid: 19823
positive_passages.text: Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$
negative_passages.docid: 93545
negative_passages.text: Showing the sequence converges $a_{1}=\frac{1}{2}$, $a_{n+1}=\frac{1}{2+a_{n}}$
negative_passages.docid: 53688
negative_passages.text: How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$?
negative_passages.docid: 89325
negative_passages.text: $a_{m+n}\leq a_ma_n$ implies the convergence of $\sqrt[n]a_n$
negative_passages.docid: 133191
negative_passages.text: Finding limit using inequalities: $\liminf \frac{a_{n+1}}{a_n} \le \liminf (a_n)^ {1/n}\le\limsup (a_n)^ {1/n}\le \limsup \frac{a_{n+1}}{a_n}$
negative_passages.docid: 126148
negative_passages.text: Convergence/divergence of $\sum\frac{a_n}{1+na_n}$ when $a_n\geq0$ and $\sum a_n$ diverges
negative_passages.docid: 155475
negative_passages.text: $\lim\limits_{n\to \infty} \int_0^1 |f(x)-a_nx-b_n| dx=0$ implies $(a_n)_n,(b_n)_n$ are convergent
negative_passages.docid: 110680
negative_passages.text: Convergence of recursive sequence $a_{n+1} =\frac{ 1}{k} \left(a_{n} + \frac{k}{a_{n}}\right)$
negative_passages.docid: 46830
negative_passages.text: Prove that the sequence $a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$ is convergent and find its limit
negative_passages.docid: 116046
negative_passages.text: Prove that if $\sum_{n=1}^\infty a_{2n}$ and $\sum_{n=1}^\infty a_{2n-1}$ converge then $\sum_{n=1}^\infty a_n$ 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: 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: 27748
negative_passages.text: Convergence of $\sum \frac{a_n}{S_n ^{1 + \epsilon}}$ where $S_n = \sum_{i = 1} ^ n a_n$
negative_passages.docid: 107399
negative_passages.text: A sequence and convergence
negative_passages.docid: 35447
negative_passages.text: If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_k $, prove that $\sum \frac{a_n}{b_n}$ diverges
negative_passages.docid: 22450
negative_passages.text: Convergence of $\sum\limits_{n=2}^\infty \frac{1}{n^\alpha \ln^\beta (n)} $ for nonnegative $\alpha$ and $\beta$
negative_passages.docid: 102581
negative_passages.text: Does $\sum_{n=1}^{\infty} \frac{1}{n^{1 + 1/n}}$ converge?