Answer to Question #340457 in Real Analysis for Yaa

Question

) Give an example to show that if the convergence of 􏰄 an is conditional and (bn) is a bounded

∞

sequence, then 􏰄 anbn may diverge.

Answer 1

ANSWER :$\sum_{n=1}^{\infty}a_{n}=\sum_ {n=1}^{\infty}\frac{(-1)^{n } }{n},\, \, b_{n}=(-1)^{n}$

EXPLANATION The series $\sum_ {n=1}^{\infty}\frac{(-1)^{n } }{n}$ is an alternating series, since the sequence $a_{n}=(-1)^{n}c_{n}$ and $c_{n}= \frac {1}{n }$ is decreasing , $\lim _{n\rightarrow\infty}c_{n}=0$ . The series $\sum_ {n=1}^{\infty}\frac{1 }{n}$ diverges, because it is a $p-$ series for $p=1$ , hence the series $\sum_ {n=1}^{\infty}\frac{(-1)^{n } }{n}$ converges conditionally . The sequence $b_{n}=(-1)^{n}$ is a bounded sequence , since $-1\leq b_{n}\leq 1$ for all $n\in\N$ ($b_{2n }=1,\, b_{2n-1}=-1$ ). Since $a_{n}\cdot b_{n}=\frac {1}{n}$ , then the series $\sum_{n=1}^{\infty}a_{n}\cdot b_{n}=\sum_ {n=1}^{\infty}\frac{1 }{n}$ diverges.