query: If $\lim_{n \to \infty }b_n=L$, and $\sum_{n=1}^\infty a_n$ is convergent, then $\sum_{n=1}^\infty a_nb_n$ is convergent.
positive_passages.docid: 119776
positive_passages.text: Prove or disprove: if $\sum\limits_{n=0}^∞a_n$ converges and $\lim\limits_{n\to\infty}b_n =0$, then $\sum\limits_{n=0}^{\infty}a_n b_n$ converges.
negative_passages.docid: 85285
negative_passages.text: Show $\sum_{i=1}^\infty x_i y_i$ is absolutely convergent
negative_passages.docid: 131927
negative_passages.text: lim inf $|a_n|=0 \implies \sum_{k=1}^\infty a_{n_k}$ converges
negative_passages.docid: 84818
negative_passages.text: Convergence of $\sum_{n=1}^{\infty}x_n\overline{y_n}$ for every $(x_n)\in c_0$ implies $\sum_{n=1}^{\infty}|y_n|<\infty$
negative_passages.docid: 1098
negative_passages.text: Does $\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}$ converge?
negative_passages.docid: 44288
negative_passages.text: Is my proof of convergence for $\sum\limits^{\infty}_{n=1} \frac {2^n} {n^n}$ true?
negative_passages.docid: 131425
negative_passages.text: Prove that the convergence of $\prod_{n=1}^{\infty} b_n$ implies $\lim_{n→∞} b_n = 1$
negative_passages.docid: 87011
negative_passages.text: Prove that for the series $\sum_{n \in \mathbb{N}}|\zeta_n\mu_n|$ to be convergent for all $\zeta \in l^p \implies \mu \in l^q$
negative_passages.docid: 104344
negative_passages.text: Prove if $|a_{n+1}|/|a_n| \leq |b_{n+1}|/|{b_n}|$ and $\sum |b_n|$ is convergent, then $\sum |a_n|$ is convergent
negative_passages.docid: 132277
negative_passages.text: $\sum a_{n}$ is convergent, $\sum a_{n}^2$ is divergent. Prove $\sum a_{n}$ is conditionally convergent.
negative_passages.docid: 5181
negative_passages.text: $(z_{2n})_{n\in \mathbb{N}}$ and $(z_{2n+1})_{n\in \mathbb{N}}$ converges to same limit. Show $(z_n)_{n\in \mathbb{N}}$ converges to the same limit.
negative_passages.docid: 7381
negative_passages.text: $\sum_{k=1}^{\infty}a_kb_k$ converges for every bounded sequence {$b_k$}, prove that $\sum_{k=1}^{\infty}a_k$ converges absolutely.
negative_passages.docid: 118271
negative_passages.text: $b_n$ bounded, $\sum a_n$ converges absolutely, then $\sum a_nb_n$ also
negative_passages.docid: 125341
negative_passages.text: Does $\sum_{n=1}^{\infty} \frac{1}{n+1}$ converge?
negative_passages.docid: 72606
negative_passages.text: Prove that : $a_n>0$, Then, $\sum_{n=1}^\infty a_n$ converges iff $\sum_{n=1}^\infty \sin(a_n)$ converges.
negative_passages.docid: 122942
negative_passages.text: Prove that if $\sum |a_{n}|$ converges then $\sum a_{n}^{2}$ converges
negative_passages.docid: 26500
negative_passages.text: Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge