Q. 6 - Calculus

Question

Explain how you could adapt the limit comparison test to analyze a series $\sum _{k=1}^{\infty }{a}_k$  in which all of the terms are negative. 

Step-by-Step Solution

Answer

To adapt limit comparison, apply it on $\sum _{k=1}^{\infty }\left(-a_k\right)$ because $-a_k$ is positive for all k

Step 1. Given information

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

Step 2. Limit comparison test

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

Now the series $\sum _{k=1}^{\infty }{a}_k$ has all terms negative, therefore $-a_k$ is positive for all k.

To adapt limit comparison, apply it on $\underset{k=1}{\overset{\infty }{\sum (-}}{a}_k)$ because $-a_k$ is positive.