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 $\frac{L}{M}$.

.

Short 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-by-step solution

Step 1. Given information: 

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

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

$$\begin{gathered} s=\frac{1}{1-{\displaystyle \frac{1}{4}}}(Sumofgeometricseries) \\ =\frac{4}{4-1}=\frac{4}{3} \\ \therefore theseries\sum _{k=0}^{\infty }{a}_k=\sum _{k=0}^{\infty }\frac{1}{4^k}isconvergetothesum\frac{4}{3} \end{gathered}$$

Step 4. Finding the convergent geometric series ∑k=0 ∞bk =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 :

$$\begin{gathered} s=\frac{1}{1-{\displaystyle \frac{1}{2}}}(Sumofgeometricseries) \\ =\frac{2}{2-1}=\frac{2}{1} \\ \therefore theseries\sum _{k=0}^{\infty }{b}_k=\sum _{k=0}^{\infty }\frac{1}{2^k}isconvergetothesum2 \end{gathered}$$

Step 6. Finding series converge to LM :

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 :

$$\begin{gathered} s=\frac{1}{1-{\displaystyle \frac{1}{2}}}(Sumofgeometricseries) \\ =\frac{2}{2-1}=\frac{2}{1} \\ \therefore theseries\sum _{k=0}^{\infty }\frac{a_k}{b_k}=\sum _{k=0}^{\infty }\frac{1}{2^k}isconvergetothesum2 \end{gathered}$$