# Sum of Infinite Geometric Series

$S = \frac{a}{1 - r}$

## Sum of Infinite Geometric Series

A Sum of Infinite Geometric Series calculator computes the sum of a series with a given first term $$a$$ and common ratio $$r$$ using the formula $$S = \frac{a}{1 – r}$$, for $$|r| < 1$$.

### Tutorial: Sum of Infinite Geometric Series

An infinite geometric series is a sum of terms in which each term is a constant multiple of the previous term. Understanding how to find the sum of such a series can be very useful in mathematics.

Formula for the Sum of an Infinite Geometric Series

The sum $$S$$ of an infinite geometric series with the first term $$a$$ and common ratio $$r$$ (where $$|r| < 1$$) is given by:

$S = \frac{a}{1 – r}$

Here:
– $$a$$ is the first term.
– $$r$$ is the common ratio.

#### Conditions

The series will only converge (i.e., have a finite sum) if the absolute value of the common ratio $$|r|$$ is less than 1.

#### Example

Let’s find the sum of the infinite geometric series:

$3 + 1.5 + 0.75 + 0.375 + \cdots$

Step 1: Identify the first term $$a$$ and the common ratio $$r$$.

– The first term $$a$$ is $$3$$.
– To find the common ratio $$r$$, divide the second term by the first term:

$r = \frac{1.5}{3} = 0.5$

Step 2: Check if $$|r| < 1$$.

– Since $$|0.5| < 1$$, the series converges, and we can use the formula.

Step 3: Use the formula to find the sum.

$S = \frac{a}{1 – r} = \frac{3}{1 – 0.5} = \frac{3}{0.5} = 6$

So, the sum of the infinite geometric series $$3 + 1.5 + 0.75 + 0.375 + \cdots$$ is:

$S = 6$

#### Summary

To find the sum of an infinite geometric series:
1. Identify the first term $$a$$ and the common ratio $$r$$.
2. Ensure that $$|r| < 1$$.
3. Use the formula $$S = \frac{a}{1 – r}$$.