A geometric series is of the following type:

being the0 and r the reason.

Ex: 1 + 2 + 4 + 8 + 16 +…

a = 1

r =

## Sum of a geometric series

The geometric series

**Converge** and have sum if | r | <1.**Diverge** if | r | 1.

## Comparison Test

**Be ** and two series of terms **positive. **So:

* If , being **"ç"** a real number then the series are **both converging** or **both divergent.**

* If what if converge then also converges.

* If what if diverges then also diverges.

NOTE: If the_{no} is expressed by a fraction, we must consider both the numerator and the denominator of b_{no} only the most important terms.

Ex: Check if the given series converges or diverges:

is a geometric series of 1/3 ratio, so it is convergent. Applying the comparison test, we have:

Therefore, it is concluded that the series converges.

Next: P-Series, Alternate Series, and Power Series