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Consider the two convergent geometric series $\underset{k=0}{\overset{∞}{∑}}{a}_k=L$and $\underset{k=0}{\overset{∞}{∑}}{b}_k=M$with all positive terms that localid="1649338303873" $\underset{k=0}{\overset{∞}{∑}}\frac{a_k}{b_k}$diverges.

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 ratio less than 1 is convergent.

Therefore, localid="1649339670370" $\underset{k=0}{\overset{∞}{∑}}{a}_k=\underset{k=0}{\overset{∞}{∑}}\frac{1}{2^k}$is convergent.

$\underset{k=0}{\overset{∞}{∑}}{2}^k$The series $\underset{k=0}{\overset{∞}{∑}}\frac{a_k}{b_k}$is

localid="1649340360408" $$\begin{gathered} \underset{k=0}{\overset{∞}{∑}}\frac{a_k}{b_k}=\underset{k=0}{\overset{∞}{∑}}\frac{4^k}{2^k} \\ =\underset{k=0}{\overset{∞}{∑}}{2}^k \end{gathered}$$

The series is$\underset{k=0}{\overset{∞}{∑}}{2}^k$a geometric series with common ratio localid="1649340245008" $r=2$, which is less than 1 . The geometric series with ratio greater than 1 is diverges.

Therefore, $\underset{k=0}{\overset{∞}{∑}}\frac{a_k}{b_k}=\underset{k=0}{\overset{∞}{∑}}{2}^k$is diverges.