Q. 5 - Calculus

Question

Provide a more general statement of the comparison test in which the inequality $0\le a_k\le b_k$ k holds only for integers $k > K$, where K is an arbitrary positive integer. Explain why your statement is valid. 

Step-by-Step Solution

Answer

The given statement is valid.

Step 1. Given information

A condition is given as $0\le a_k\le b_k$

Step 2. Verification

The series $\sum _{k=1}^{\infty }{b}_k$ is converges therefore the series $\sum _{k=1}^{\infty }{a}_k$ is also converges according to the comparison test. Because the condition of convergence or divergence is depend on the tails of the series.

Therefore the given statement is valid.