Q. 13 - Calculus

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. 

Step-by-Step Solution

Answer

Hence proved.

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 $\sum _{k=1}^{\infty }\left(-a_k\right)$.