Question

The Ratio Test: Let $\sum _{k=1}^{\infty }{a}_k$be a series with nonnegative terms, and let $p=\underset{k\to \infty }{\lim }\_\_\_$.

If $p<1$, the series ___.

If $p>1$, the series ___.

If $p=1$, ___.

Short answer

Let $\sum _{k=1}^{\infty }{a}_k$be a series with nonnegative terms, and let localid="1649361444772" $p=\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}$.

If $p<1$, the series converges.

If $p>1$, the series diverges.

If $p=1$, the test is inconclusive.

Step-by-step solution

Step 1. Given Information

The given test is the ratio test.

Step 2. Explanation

• The ratio test is used in the case of geometric series or series with terms like $a^n,n!etc$.
• To perform the ratio test, find
• If $p<1$, the series is convergent.
• If $p>1$, the series is divergent.
• But if the limit is 1, the test cannot guarantee any behavior.