91Ó°ÊÓ

Consider the two divergent geometric series$\underset{k=0}{\overset{∞}{∑}}{a}_k$ and $\underset{k=0}{\overset{∞}{∑}}{b}_k$such that $\underset{k=0}{\overset{∞}{∑}}\left(a_k+b_k\right)$ converge.

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

The series $\underset{k=0}{\overset{∞}{∑}}1$is a geometric series with common ratio $r=1$, which is equal to 1 . The geometric series with ratio equal to 1 is divergent.

Therefore, $\underset{k=0}{\overset{∞}{∑}}{a}_k=\underset{k=0}{\overset{∞}{∑}}1$is divergent.

Consider the geometric series $\underset{k=0}{\overset{∞}{∑}}{b}_k=\underset{k=0}{\overset{∞}{∑}}(-1)$.

The series $\underset{k=0}{\overset{∞}{∑}}{b}_k=\underset{k=0}{\overset{∞}{∑}}(-1)$is a geometric series with common ratio $r=1$, which is equal to 1 .

The geometric series with ratio equal to 1 is divergent.

Therefore, localid="1649329293676" $\underset{k=0}{\overset{∞}{∑}}{b}_k=\underset{k=0}{\overset{∞}{∑}}(-1)$is divergent.

The series $\underset{k=0}{\overset{∞}{∑}}\left(a_k+b_k\right)$is

$$\begin{gathered} \underset{k=0}{\overset{∞}{∑}}\left(a_k+b_k\right)=\underset{k=0}{\overset{∞}{∑}}\left(1+(-1)\right) \\ \underset{k=0}{\overset{∞}{∑}}\left(0\right)=0 \end{gathered}$$

The partial sum of series $=\underset{k=0}{\overset{∞}{∑}}0$is a constant and hence, is convergent. Therefore, $\underset{k=0}{\overset{∞}{∑}}\left(a_k+b_k\right)=\underset{k=0}{\overset{∞}{∑}}0$is convergent.