Learn how to solve the Infinite Geometric Series using the following step-by-step guide and examples.

## Related Topics

- How to Solve Finite Geometric Series
- How to Solve Geometric Sequences
- How to Solve Arithmetic Sequences

## Step by step guide to solve Infinite Geometric Series

- Infinite Geometric Series: The sum of a geometric series is infinite when the absolute value of the ratio is more than \(1\).
- Infinite Geometric Series formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}}\)

### Infinite Geometric Series – Example 1:

Evaluate infinite geometric series described. \(S= \sum_{i=1}^ \infty 9^{i-1}\)

**Solution:**

Use this formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{i=1}^ \infty 9^{i-1}=\frac{1}{1-9}=\frac{1}{-8}=-\frac{1}{8}\)

### Infinite Geometric Series – Example 2:

Evaluate infinite geometric series described. \(S= \sum_{k=1}^ \infty (\frac{1}{4})^{k-1}\)

**Solution**:

Use this formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{k=1}^ \infty (\frac{1}{4})^{k-1}=\frac{1}{1-\frac{1}{4}}=\frac{1}{\frac{3}{4}}=\frac{4}{3}\)

### Infinite Geometric Series – Example 3:

Evaluate infinite geometric series described. \(S= \sum_{i=1}^ \infty 8^{i-1}\)

**Solution**:

Use this formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{i=1}^ \infty 8^{i-1}=\frac{1}{1-8}=\frac{1}{-7}=-\frac{1}{7}\)

### Infinite Geometric Series – Example 4:

Evaluate infinite geometric series described. \(S= \sum_{k=1}^ \infty (\frac{1}{2})^{k-1}\)

**Solution**:

Use this formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{k=1}^ \infty (\frac{1}{2})^{k-1}=\frac{1}{1-\frac{1}{2}}=\frac{1}{\frac{1}{2}}=2\)

## Exercises for Solving Infinite Geometric Series

### Determine if each geometric series converges or diverges.

- \(\color{blue}{a_{1} = –3, r = 4}\)
- \(\color{blue}{a_{1}= 5.5, r = 0.5}\)
- \(\color{blue}{a_{1} = –1, r = 3}\)
- \(\color{blue}{81 + 27 + 9 + 3 …,}\)
- \(\color{blue}{–3 + \frac{12}{5} – \frac{48}{25} + \frac{192}{125} …,}\)
- \(\color{blue}{\frac{128}{3125} – \frac{64}{625} + \frac{32}{125} – \frac{16}{25 }…,}\)

## Answers

- \(\color{blue}{Diverges}\)
- \(\color{blue}{Converges}\)
- \(\color{blue}{Diverges}\)
- \(\color{blue}{Converges}\)
- \(\color{blue}{Converges}\)
- \(\color{blue}{Diverges}\)