Q. 56 - 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 $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$and $\sum _{k=1}^{\infty }{b}_k$diverges, then $\sum _{k=1}^{\infty }{a}_k$diverges.

Step-by-Step Solution

Answer

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

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

Step 1. Given information.

Series $\sum _{k=1}^{\infty }{a}_k \& \sum _{k=1}^{\infty }{b}_k$are two series with positive terms.

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

Step 2. Proof.

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

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