query: Convergence of the series $\sum_n\sqrt{a_n\cdot a_{n+1}}$ given that $\sum_na_n$ converges
positive_passages.docid: 63900
positive_passages.text: Convergent or divergent series: $\sum_{n=1}^{\infty} \sqrt{a_n \cdot a_{n+1}}$?
negative_passages.docid: 18172
negative_passages.text: If $\sum{a_n}$ converges then does $\sum{(a_n)^2}$ converge where $a_n \ge 0$ for all $n$
negative_passages.docid: 20142
negative_passages.text: $\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$
negative_passages.docid: 16970
negative_passages.text: Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
negative_passages.docid: 75459
negative_passages.text: study the convergence of $\sum_{n=1}^\infty \log \frac{n+1}{n}$
negative_passages.docid: 109807
negative_passages.text: For what values of $a$ the series $\sum \limits_{n=1}^{\infty} \frac{a^n}{n^\sqrt n}$ converges?
negative_passages.docid: 139971
negative_passages.text: If $\sum{a_n}$ converges does that imply that $\sum{\frac{a_n}{n}}$ converges?
negative_passages.docid: 140220
negative_passages.text: Convergence of $\sum_{n=2}^{\infty}\frac{\sqrt{a_{n}}}{\ln\, n}(n^{a_{n}}-1)$
negative_passages.docid: 143263
negative_passages.text: If $\sum a_n$ is a convergent series with $S = \lim s_n$, where $s_n$ is the nth partial sum, then $\lim_{n \to \infty} \frac{s_1+...+s_n}{n} = S$
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: 35885
negative_passages.text: Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$
negative_passages.docid: 30471
negative_passages.text: Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges.
negative_passages.docid: 28587
negative_passages.text: Show that $\sum\limits_{n=1}^\infty \frac {\sqrt{a_n}}{n}$ converges if $\sum\limits_{n=1}^\infty{a_n}$ does provided that $a_n>0$
negative_passages.docid: 10562
negative_passages.text: Will $\sum^{\infty}_{n=1}|a_n|^p$ converge?
negative_passages.docid: 57372
negative_passages.text: Series convergence of $\sum_{n=1}^{\infty}\frac{1}{n^3}$
negative_passages.docid: 39302
negative_passages.text: Radius of convergence of a sum of power series
negative_passages.docid: 47357
negative_passages.text: Does the series $\sum_{n=1}^{\infty}|x|^\sqrt n$ converge pointwise? If it then what would be the sum?