Q. 17 - Calculus

Question

Find two convergent geometric series $\sum _{k=0}^{\infty }{a}_k=L$ and  $\sum _{k=0}^{\infty }{b}_k=M$ such that the series    $\sum _{k=0}^{\infty }\left(a_k.b_k\right)$ converges. Does this series converge to LM?

Step-by-Step Solution

Answer

Ans:

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

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

part (c). The serise  is converge $\sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }\frac{1}{2^k}$

part (d). The series $\sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }\frac{1}{2^k}$ converge to the sum $\frac{4}{3}$ 

              The series $\sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }\frac{1}{2^k}$ do not converge to the sum of $\sum _{k=0}^{\infty }{a}_k·\sum _{k=0}^{\infty }{b}_k$ .

Step 1. Given information:

Consider the two convergent geometric series $\sum _{k=0}^{\infty }{a}_k=L$ and $\sum _{k=0}^{\infty }{b}_k=M$ such that $\sum _{k=0}^{\infty }{a}_k·b_k$ converge.

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

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 a 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 = M

Consider the geometric series $\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }\frac{1}{2^k}$.

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

Step 4. finding the serise ∑ k = 0 ∞ a k · b k is converges or not:

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

$$\begin{gathered} \sum _{k=0}^{\infty }{a}_k.b_{k }=\sum _{k=0}^{\infty }\frac{1}{2^k}\times \frac{1}{2^k} \\ =\sum _{k=0}^{\infty }\frac{1}{4^k} \end{gathered}$$

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

Step 5. Finding series converge to LM :

The serles $\sum _{k=0}^{\infty }{a}_k·b_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$ Is convergent with ratio $r=\frac{1}{4}$ and converge to the sum:

$S=\frac{1}{1-\frac{1}{4}}\left(S_{\infty }=\frac{a}{1-r}\right)$

$\frac{4}{4-1}=\frac{4}{3}$

Therefore, the series $\sum _{k=0}^{\infty }{a}_k·b_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$ converge to the sum $\frac{4}{3}$.

Hence, the series $\sum _{k=0}^{\infty }{a}_k·b_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$ do not converge to the sum of $\sum _{k=0}^{\infty }{a}_k·\sum _{k=0}^{\infty }{b}_k$.