Question

Find two divergent series $\sum _{k=1}^{\infty }{a}_k$and $\sum _{k=1}^{\infty }{b}_k$such that the series $\sum _{k=1}^{\infty }(a_k+b_k)$converges. Carefully use the sequence of partial sums for this new series to show that your answer is correct.

Short answer

Ans:

part (a). divergent series of $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }1$

part (b). divergent series of $\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }(-1)$

part (c).The partial sum of series $=\sum _{k=0}^{\infty }0$is a constant and hence, is convergent. Therefore, $\sum _{k=0}^{\infty }\left(a_k+b_k\right)=\sum _{k=0}^{\infty }0$is convergent.

Step-by-step solution

Step 1. Given information:

Consider the two divergent geometric series$\sum _{k=0}^{\infty }{a}_k$ and $\sum _{k=0}^{\infty }{b}_k$such that $\sum _{k=0}^{\infty }\left(a_k+b_k\right)$ converge.

Step 2. Finding divergent series of ∑k=0∞ak :

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

The series $\sum _{k=0}^{\infty }1$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, $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }1$is divergent.

Step 3. Finding divergent series of ∑k=0∞bk :

Consider the geometric series $\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }(-1)$.

The series $\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }(-1)$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" $\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }(-1)$is divergent.

Step 4. Using the sequence of partial sums in new series :

The series $\sum _{k=0}^{\infty }\left(a_k+b_k\right)$is

$$\begin{gathered} \sum _{k=0}^{\infty }\left(a_k+b_k\right)=\sum _{k=0}^{\infty }\left(1+(-1)\right) \\ \sum _{k=0}^{\infty }\left(0\right)=0 \end{gathered}$$

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