Question

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

If ρ < 1, the series ___.

If ρ > 1, the series ___.

If ρ = 1, ___.

Short answer

Let $\sum _{k=1}^{\infty }{a}_k$be a series with nonnegative terms, and let localid="1649361486379" $p=\underset{k\to \infty }{\lim }\sqrt[k]{a_k}$.

If $p<1$, the series diverges.

If $p>1$, the series converges.

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

Step-by-step solution

Step 1. Given Information

The given test is the root test.

Step 2. Explanation

• To perform the root test, find$p=\underset{k\to \infty }{\lim }\sqrt[k]{a_k}$
• 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.