Q. 9 - Calculus

Question

If you suspect that a series $\sum _{k=1}^{\infty }{a}_k$ converges, explain why you would want to compare the series with a convergent series, using either the comparison test or the limit comparison test. 

Step-by-Step Solution

Answer

If the series $\sum _{k=1}^{\infty }{a}_k$ is convergent, comparing it to divergent series will yield no results since the behaviour of the series $\sum _{k=1}^{\infty }{a}_k$is dependent on the behaviour of the series $\sum _{k=1}^{\infty }{b}_k$.

As a result, if the $\sum _{k=1}^{\infty }{a}_k$series converges, it must be compared to a convergent series.

Step 1. Given information

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

Step 2. Verification

The limit comparison test for $\sum _{k=1}^{\infty }{a}_k$ and $\sum _{k=1}^{\infty }{b}_k$ are the series having positive terms the the following conditions may apply,

If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=L$, L must be positive number then it may be either converging or diverging.

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

If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$then if $\sum _{k=1}^{\infty }{b}_k$ diverges $\sum _{k=1}^{\infty }{a}_k$ diverges

If the series $\sum _{k=1}^{\infty }{a}_k$ is convergent, comparing it to divergent series will yield no results since the behaviour of the series $\sum _{k=1}^{\infty }{a}_k$ is dependent on the behaviour of the series $\sum _{k=1}^{\infty }{b}_k$.

As a result, if the $\sum _{k=1}^{\infty }{a}_k$ series converges, it must be compared to a convergent series.