Q. 7 - Calculus

Question

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

Step-by-Step Solution

Answer

Hence explained.

Step 1. Given information.

The given series is $\sum _{k=1}^{\infty }{a}_k$.

Step 2. Explanation.

The ratio test is given by:

$$\begin{gathered} \text{ 1. If }L<1\text{ series converges. } \\ \text{ 2. If }L>1\text{ series diverges. } \\ \text{ 3. If }L=1\text{ the test is inconclusive. } \\ where L=\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k} and \sum _{k=1}^{\infty }{a}_k is the series. \end{gathered}$$

Now, 

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