Sum of Infinite Geometric Series

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 \( S = \frac{a}{1 – r} \).