# Infinite Geometric Series

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

## 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: $$\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}}$$

### Example 1:

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

Solution:

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

### Example 2:

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

Solution:

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

### Example 3:

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

Solution:

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

### 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

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

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

• $$\color{blue}{Diverges}$$
• $$\color{blue}{Converges}$$
• $$\color{blue}{Diverges}$$
• $$\color{blue}{Converges}$$
• $$\color{blue}{Converges}$$
• $$\color{blue}{Diverges}$$

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