# Science:Infinite Series Module/Units/Unit 1/1.3 Infinite Series/1.3.04 Geometric Series Example

## Example

Determine if the following series is convergent, and if so, find its sum:

## Complete Solution

The sum of the series is therefore 3/5.

## Explanation of Each Step

### Step (1)

We first rewrite the problem so that the summation starts at one and is in the familiar form of a geometric series, whose general form is

After bringing the negative one and the three fifths together, we see that our given infinite series is geometric with common ratio -3/5.

For a geometric series to be convergent, its common ratio must be between -1 and +1, which it is, and so our infinite series is convergent.

We must now compute its sum.

### Step (2)

The given series

starts the summation at , so we shift the index of summation by one:

Our sum is now in the form of a geometric series with *a* = 1, *r* = -2/3. Since |*r*| < 1, the series converges, and its sum is

### Step (3)

In Step (3) we applied the formula for the sum of a geometric series:

This formula was derived in a previous section of this lesson.

## Possible Challenge Areas

### Step (2)

Students who may have been confused by this step may wish to refer to the previous lesson on Sigma Notation, where this process was explained in more detail.