Question

Explain how you could adapt the root test to analyze a series$\sum _{k=1}^{\infty }{a}_k$in which the terms of the series are all negative.

Short answer

Hence proved.

Step-by-step solution

Step 1. Given information.

We are given$\sum _{k=1}^{\infty }{a}_k$.

Step 2. Explanation.

According to the root test,

$$\begin{gathered} \text{1. If}L<1\text{series converges.} \\ \text{2. If}L>1\text{series diverges.} \\ \text{3. If}L=1\text{the test is inconclusive.} \end{gathered}$$

Where, Series is $\sum _{k=1}^{\infty }{a}_k$and $L=\underset{k\to \infty }{\lim }{\left(a_k\right)}^{\frac{1}{k}}$.

But if the series has all the terms negative, then negative sign can be adjusted and Root test can be used on the series $\sum _{k=1}^{\infty }\left(-a_k\right)$.