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 .

Short answer

Hence proved.

Step-by-step solution

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 $\sum _{k=1}^{\infty }{a}_{k}seriesandL=\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}$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.