Find two convergent geometric series $\sum _{k=0}^{\infty }{a}_k=L$and $\sum _{k=0}^{\infty }{b}_k=M$such that the series $\sum _{k=0}^{\infty }\left(a_k.b_k\right)$converges. Does this series converge to LM?

Consider the two convergent geometric series $\sum _{k=0}^{\infty }{a}_k=L$and $\sum _{k=0}^{\infty }{b}_k=M$such that $\sum _{k=0}^{\infty }{a}_k·b_k$ converge.

Consider the geometric series $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{2^k}$.

The series $\sum _{k=0}^{\infty }\frac{1}{2^k}$is a geometric series with common ratio $r=\frac{1}{2}$, which is less than 1 .

The geometric series with a ratio less than 1 is convergent.

Therefore, $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{2^k}$is convergent.

Consider the geometric series $\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }\frac{1}{2^k}$.

The series $\sum _{k=0}^{\infty }\frac{1}{2^k}$is a geometric series with common ratio $r=\frac{1}{2}$, which is less than 1 . The geometric series with ratio less than 1 is convergent.

Therefore, $\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }\frac{1}{2^k}$is convergent.

The series $\sum _{k=0}^{\infty }{a}_k.b_k$is

$$\begin{gathered} \sum _{k=0}^{\infty }{a}_k.b_k=\sum _{k=0}^{\infty }\frac{1}{2^k}\times \frac{1}{2^k} \\ =\sum _{k=0}^{\infty }\frac{1}{4^k} \end{gathered}$$

The series $\sum _{x=0}^{\infty }\frac{1}{4^k}$is a geometric series with common ratio $r=\frac{1}{4}$, which is less than 1 .

The geometric series with ratio less than 1 is convergent.

Therefore, localid="1649331930674" $\sum _{k=0}^{\infty }{a}_k·b_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$is convergent.

The serles $\sum _{k=0}^{\infty }{a}_k·b_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$Is convergent with ratio $r=\frac{1}{4}$and converge to the sum:

$S=\frac{1}{1-\frac{1}{4}}\left(S_{\infty }=\frac{a}{1-r}\right)$

$\frac{4}{4-1}=\frac{4}{3}$

Therefore, the series $\sum _{k=0}^{\infty }{a}_k·b_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$converge to the sum $\frac{4}{3}$.

Hence, the series $\sum _{k=0}^{\infty }{a}_k·b_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$do not converge to the sum of $\sum _{k=0}^{\infty }{a}_k·\sum _{k=0}^{\infty }{b}_k$.