query: Convergence of $a_n$ to $(a_1a_2)^{\frac1{3}}$
positive_passages.docid: 29266
positive_passages.text: Limit of a sequence defined by recursive relation : $ a_n = \sqrt{a_{n-1}a_{n-2}}$
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: 41445
negative_passages.text: Show that the sequence $a_n\leq a_{2n}+a_{2n+1}$ 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: 135110
negative_passages.text: Pointwise convergence of $h_n(x) = x^{1+\frac{1}{2n-1}}$
negative_passages.docid: 9263
negative_passages.text: Convergence of $\sum_{n=1}^\infty\frac{n!\, i^n}{n^n}$
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: 145863
negative_passages.text: sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
negative_passages.docid: 33495
negative_passages.text: A limit problem about $a_{n+1}=a_n+\frac{n}{a_n}$
negative_passages.docid: 32423
negative_passages.text: Convergence of $\sum _{k=1}^\infty (1-\frac{1}{k})^{k^2}$
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: 85268
negative_passages.text: Convergence of $\sum_{n=1}^{\infty}{\sqrt[n]{n}-1 \over n}$
negative_passages.docid: 51694
negative_passages.text: Radius of convergence of a power series with $a_n$ convergent
negative_passages.docid: 12031
negative_passages.text: Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
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: 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: 20136
negative_passages.text: Convergence of the sequence $ \sqrt {n-2\sqrt n} - \sqrt n $