Q. 3 - Calculus

Question

Explain how you could adapt the 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

We can apply comparison test on the series $\sum _{k=1}^n{a}_k$

Step 1. Given information

An general term of an series is given as $\sum _{k=1}^{\infty }{a}_k$

Step 2. Verification

In comparison test $\sum _{k=1}^{\infty }{a}_k$ and $\sum _{k=1}^{\infty }{b}_k$ be two series with positive terms with the condition as $0\le a_k\le b_k$ for every value of k. If the series $\sum _{k=1}^n{b}_k$ converges then the series $\sum _{k=1}^n{a}_k$ converges.

In the series $\sum _{k=1}^n{a}_k$, all terms are negative. Thus, for all values of k, $a_k$ is negative. Therefore, $-a_k$ is positive for all values of k.

Thus we can apply comparison test on the series $\sum _{k=1}^n{a}_k$