Question

What is a geometric series? What determines the convergence of a geometric series?

Short answer

The sum of the terms in geometric progression gives geometric series.

The convergence of the geometric progression depends upon the common ratio.

Step-by-step solution

Step 1. Given information

The series which is obtained by multiplying the terms by a fixed number; the series following is in geometric progression :

$a,ar,a{r}^2,....$

Step 2. Geometric Series

The sum of the terms in geometric progression gives geometric series.

The geometric series is written as $\underset{k=0}{\overset{\infty }{\sum a{r}^k}}$.

Step 3. Convergence of geometric series

The convergence of the geometric progression depends upon the common ratio.

If the common ratio is less than one, the series is convergent.

If the common ratio is greater than one, the series is divergent.