91Ó°ÊÓ

Consider the two convergent geometric series $\underset{k=0}{\overset{∞}{∑}}{a}_k=L$and $\underset{k=0}{\overset{∞}{∑}}{b}_k=M$such that $\underset{k=0}{\overset{∞}{∑}}{a}_k·b_k$ converge.

Consider the geometric series $\underset{k=0}{\overset{∞}{∑}}{a}_k=\underset{k=0}{\overset{∞}{∑}}\frac{1}{2^k}$.

The series $\underset{k=0}{\overset{∞}{∑}}\frac{1}{2^k}$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, $\underset{k=0}{\overset{∞}{∑}}{a}_k=\underset{k=0}{\overset{∞}{∑}}\frac{1}{2^k}$is convergent.

Consider the geometric series $\underset{k=0}{\overset{∞}{∑}}{b}_k=\underset{k=0}{\overset{∞}{∑}}\frac{1}{2^k}$.

The series $\underset{k=0}{\overset{∞}{∑}}\frac{1}{2^k}$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, $\underset{k=0}{\overset{∞}{∑}}{b}_k=\underset{k=0}{\overset{∞}{∑}}\frac{1}{2^k}$is convergent.

The series $\underset{k=0}{\overset{∞}{∑}}{a}_k.b_k$is

$$\begin{gathered} \underset{k=0}{\overset{∞}{∑}}{a}_k.b_k=\underset{k=0}{\overset{∞}{∑}}\frac{1}{2^k}×\frac{1}{2^k} \\ =\underset{k=0}{\overset{∞}{∑}}\frac{1}{4^k} \end{gathered}$$

The series $\underset{x=0}{\overset{∞}{∑}}\frac{1}{4^k}$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" $\underset{k=0}{\overset{∞}{∑}}{a}_k·b_k=\underset{k=0}{\overset{∞}{∑}}\frac{1}{4^k}$is convergent.

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

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

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

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

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