Q. 18 - 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 $\sum _{k=0 }^{\infty }\frac{a_{k }}{b_k}$ bk converges but whose sum is not $\frac{L}{M}$.

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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 2 

              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  $\frac{\sum _{k=0 }^{\infty }{a}_k}{\sum _{k=0}^{\infty } b_k}$.

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 $\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{4^k}$.

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 3. Checking the convergent geometric series :

$$\begin{gathered} s=\frac{1}{1-{\displaystyle \frac{1}{4}}}(Sum of geometric series) \\ =\frac{4}{4-1}=\frac{4}{3} \\ \therefore the series \sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{4^k} is converge to the sum\frac{4}{3} \end{gathered}$$

Step 4. 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 6. Checking the convergent geometric series :

$$\begin{gathered} s=\frac{1}{1-{\displaystyle \frac{1}{2}}}(Sum of geometric series) \\ =\frac{2}{2-1}=\frac{2}{1} \\ \therefore the series \sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }\frac{1}{2^k} is converge to the sum 2 \end{gathered}$$

Step 6. Finding series converge to L M :

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

$$\begin{gathered} \sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }\frac{2^k}{4^k} \\ =\sum _{k=0}^{\infty }\frac{1}{2^k} \end{gathered}$$

The series $\sum _{k=0}^{\infty }\frac{1}{4^k}$ 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, $\sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }\frac{1}{2^k}$ is convergent.

Step 7. Checking the convergent geometric series :

$$\begin{gathered} s=\frac{1}{1-{\displaystyle \frac{1}{2}}}(Sum of geometric series) \\ =\frac{2}{2-1}=\frac{2}{1} \\ \therefore the series \sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }\frac{1}{2^k} is converge to the sum 2 \end{gathered}$$