# Infinite geometric series

### Infinite geometric series

We know how to deal with geometric series with finite terms. But what if we have a series of infinite number of terms? In this section, we will explore the concept behind infinite geometric series, as well as learning how to determine whether an infinite geometric series is convergent or divergent. Just like all other series, we have a formula to help us find the sum to infinity.

#### Lessons

• Introduction
If $-1 < r < 1$, an infinite series is convergent:
$S_{\infty} =$ $\large\frac{t_1}{1-r}$

Otherwise, an infinite series is divergent:
$S_{\infty} =$ undefined

• 1.
Using the common ratio to determine whether a sum to infinity exists
For each geometric series determine the:
i) common ratio.
ii) sum of the first 10 terms.
iii) sum to infinity.
a)
4 + 2 + 1 + …

b)
2 - 10 + 50 - …

c)
$-\frac{9}{2}$+ 3 - 2 + …

• 2.
Expressing a repeating decimal as an infinite geometric series
For the repeating decimal:
i) express it as an infinite geometric series.
ii) write it as a fraction by evaluating the sum of the infinite geometric series.
a)
$0.\overline{461}$

b)
$5.1\overline{23}$