Question

Find two convergent geometric series $\sum _{k=0}^{\infty }{a}_k=L$and $\sum _{k=0}^{\infty }{b}_k=M$with all positive terms such that $\sum _{k=0}^{\infty }\frac{a_k}{b_k}$diverges.

Short answer

Ans:

part (a). The convergent geometric series $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{2^k}$

part (b). The convergent geometric series $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$

part (c). The series is diverges $\sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }{2}^k$

Step-by-step solution

Step 1. Given information:

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

Step 2. Finding the convergent  geometric series  ∑k=0∞ak

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 ratio less than 1 is convergent.

Therefore, localid="1649339670370" $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{2^k}$is convergent.

Step 3. Finding the convergent  geometric series  ∑k=0∞bk

Consider the geometric series $\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$.

The series $\sum _{k=0}^{\infty }\frac{1}{4^k}$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, $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$is convergent.

Step 4. Finding the converges geometric series of ∑k=0∞akbk with all positive terms:

$\sum _{k=0}^{\infty }{2}^k$The series $\sum _{k=0}^{\infty }\frac{a_k}{b_k}$is

localid="1649340360408" $$\begin{gathered} \sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }\frac{4^k}{2^k} \\ =\sum _{k=0}^{\infty }{2}^k \end{gathered}$$

The series is$\sum _{k=0}^{\infty }{2}^k$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, $\sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }{2}^k$is diverges.