Question

The Limit Comparison Test: Let $\sum _{k=1}^{\infty }{a}_k$and $\sum _{k=1}^{\infty }{b}_k$be two series with positive terms.

If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=L$, where L is ___, then either the series both converge or both diverge.

If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$and $\sum _{k=1}^{\infty }{b}_k$___, then $\sum _{k=1}^{\infty }{a}_k$___.

(There are two correct ways to fill in the last two blanks.)

Short answer

Let $\sum _{k=1}^{\infty }{a}_k$and $\sum _{k=1}^{\infty }{b}_k$be two series with positive terms. If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=L$, where L is a finite positive number, then either the series both converge or both diverge.

If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$and role="math" localid="1649359537559" $\sum _{k=1}^{\infty }{b}_k$converges, then $\sum _{k=1}^{\infty }{a}_k$converges.

Step-by-step solution

Step 1. Given Information

The given test is the limit comparison test.

Step 2. Explanation

• If there are two series such that $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=L$then:
• If $L=0$and $\sum _{k=1}^{\infty }{b}_k$converges, then $\sum _{k=1}^{\infty }{a}_k$also converges.
• If $L=\infty$and $\sum _{k=1}^{\infty }{b}_k$diverges , then $\sum _{k=1}^{\infty }{a}_k$also diverges.