How to Solve Infinite Geometric Series? (+FREE Worksheet!)

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

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Infinite Geometric Series – Example 2:

Evaluate the 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 the 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 the 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

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