Q. 20 - Calculus

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.

Step-by-Step Solution

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 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 $\sum _{k=0}^{\infty }\frac{a_k}{b_k}$diverges.

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

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

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

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 ∞ a k b k with all positive terms:

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

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

The series is$\sum _{k=0}^{\infty }{2}^k$ a geometric series with common ratio $r=2$, 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.