Q. 57 - Calculus

Question

In Exercises 56 and 57 we ask you to complete the proof of Theorem 7.33. For these exercises let $\sum _{k=1}^{\infty }{a}_k$and $\sum _{k=1}^{\infty }{b}_k$be two series with positive terms.

Show that if $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$and $\sum _{k=1}^{\infty }{b}_k$converges, then $\sum _{k=1}^{\infty }{a}_k$converges.

Step-by-Step Solution

Answer

As $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$so a positive integer N also exists such that $0<a_k<b_k$for all $k>N.$

According to The Comparison Test, if $\sum _{k=1}^{\infty }{a}_k \& \sum _{k=1}^{\infty }{b}_k$two series with nonnegative terms where $0<a_k<b_k$and $\sum _{k=1}^{\infty }{b}_k$converge, then the series also $\sum _{k=1}^{\infty }{a}_k$converges.

Step 1. Given Information.

The given series is $\sum _{k=1}^{\infty }{a}_k .$

series $\sum _{k=1}^{\infty }{b}_k$converses and $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0.$

Step 2. Proof.

As given $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$so a positive integer N also exists such that $0<a_k<b_k$for all $k>N.$

According to The Comparison Test, if $\sum _{k=1}^{\infty }{a}_k \& \sum _{k=1}^{\infty }{b}_k$two series with nonnegative terms where $0<a_k<b_k$and$\sum _{k=1}^{\infty }{b}_k$ converge, then the series $\sum _{k=1}^{\infty }{a}_k$also converges.