Q. 8 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1. Given information

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

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 then $\sum _{k=1}^{\infty }{a}_k$ converges

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

If the series $\sum _{k=1}^{\infty }{a}_k$ is divergent, comparing it to convergent 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 diverges, it must be compared to a divergent series.