Question

Given a series $\sum _{k=1}^{\infty }{a}_k$, in general the divergence test is inconclusive when . For a $a_k\to 0$geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

Short answer

Geometric series are convergent only when the ratio holds $\left|r\right|<1$.

Step-by-step solution

Step 1. Given Information.

The series:

$$\begin{gathered} \sum _{k=1}^{\infty }{a}_k \\ {a}_k\to 0 \end{gathered}$$

Step 2. Consider the series.

Consider the geometric series,

$\sum _{k=1}^{\infty }c{r}^4$

The value of the series is zero only when the ratio is less than one.

Step 3. Geometric series are convergent.

Geometric series are convergent only when the ratio holds $\left|r\right|<1$.

So, for a geometric series, if the limit of the terms of the series is zero, the series converges.