Question

The Comparison Test: Let $\sum _{k=1}^{\infty }{a}_k\mathrm{and}\sum _{k=1}^{\infty }{b}_k$be two series with ___ terms such that $0\_\_\_a_k\_\_\_b_k$for every positive integer k. If the series $\sum _{k=1}^{\infty }{b}_k$___ , then the series $\sum _{k=1}^{\infty }{a}_k$___.

Short answer

Let $\sum _{k=1}^{\infty }{a}_k\mathrm{and}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{b}}_{\mathrm{k}}$be two series with positive terms such that $0\le a_k\le b_k$for every positive integer k. If the series $\sum _{k=1}^{\infty }{a}_k$diverges, then the series $\sum _{k=1}^{\infty }{b}_k$ diverges.

Step-by-step solution

Step 1. Given Information.

The given test is the comparison test.

Step 2. Explanation

• A comparison test is used if there are 2 positive series such that $0\le a_k\le b_k$.
• If such series are found, the if $\sum _{k=1}^{\infty }{a}_k$diverges, $\sum _{k=1}^{\infty }{b}_k$also diverges.
• And if $\sum _{k=1}^{\infty }{b}_k$converges, $\sum _{k=1}^{\infty }{a}_k$also converges.