Q. 55 - Calculus

Question

Prove that if $\sum _{k=1}^{\infty }{a}_k$and $\sum _{k=1}^{\infty }{b}_k$are two convergent series with $a_k \ge 0$and $b_k\ge 0$for every positive integer k, then the series $\sum _{k=1}^{\infty }\left(a_k·b_k\right)$converges.

Step-by-Step Solution

Answer

$\sum _{k=1}^{\infty }{b}_k$is convergent, so according to the divergence test, $\underset{k\to \infty }{\lim }{b}_k=0.$

Then

$$\begin{gathered} \underset{k\to \infty }{\lim }\frac{a_k·b_k}{a_k}=\underset{k\to \infty }{\lim } b_k \\ \underset{k\to \infty }{\lim }\frac{a_k·b_k}{a_k}=0 \end{gathered}$$

According to The Limit Comparison Test, if $\underset{k\to \infty }{\lim }\frac{a_k·b_k}{a_k}=0$and $\sum _{k=1}^{\infty }{b}_k$converges, then $\sum _{k=1}^{\infty }\left(a_k·b_k\right)$also converges.

Step 1. Given Information.

Series $\sum _{k=1}^{\infty }{a}_k \& \sum _{k=1}^{\infty }{b}_k\mathbf{ }$is a convergent where $a_k\ge 0 \& b_k\ge 0$for every positive integer k.

The given series are the following.

$\sum _{k=1}^{\infty }\left(a_k·b_k\right)$

Step 2. Proof

Consider $\underset{k\to \infty }{\lim }\frac{a_k·b_k}{a_k}=\underset{k\to \infty }{\lim } b_k.$

 $\sum _{k=1}^{\infty }{b}_k$is convergent, so according to the divergence test, $\underset{k\to \infty }{\lim }{b}_k=0.$

So $$\begin{gathered} \underset{k\to \infty }{\lim }\frac{a_k·b_k}{a_k}=\underset{k\to \infty }{\lim } b_k \\ \underset{k\to \infty }{\lim }\frac{a_k·b_k}{a_k}=0 \end{gathered}$$

According to The Limit Comparison Test, if $\underset{k\to \infty }{\lim }\frac{a_k·b_k}{a_k}=0$and $\sum _{k=1}^{\infty }{b}_k$converges, then $\sum _{k=1}^{\infty }\left(a_k·b_k\right)$converges.