Q. 9 - Calculus

Question

Let $\sum _{k=1}^{\infty }{a}_k$ be a series in which all the terms are positive. If $\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}>1$, explain why both the ratio test and the divergence test could be used to show that the series diverges .

Step-by-Step Solution

Answer

Hence proved.

Step 1. Given information.

We are given,

$\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}>1$

Step 2. Ratio Test.

The ratio test for $\sum _{k=1}^{\infty }{a}_k series and L=\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}$ is given by,

$$\begin{gathered} \text{ 1. If }L<1\text{ series converges. } \\ \text{ 2. If }L>1\text{ series diverges. } \\ \text{ 3. If }L=1\text{ the test is inconclusive. } \\ \text{ Since, }\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}>1\text{, therefore, }L>1\text{. } \end{gathered}$$

Hence, it diverges.

Step 2. Divergence Test.

We know,

According to the Divergence Test if the sequence $\left\{a_k\right\}$ does not converge to zero, then the series $\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}>1$ diverges.

Since $\underset{k\to \infty }{\lim }{a}_k\ne 0$

Here, $a_k$ does not converge to 0 .

Hence, the series$\sum _{k=1}^{\infty }{a}_k$ diverges by Divergence Test.