Previous: Power Series Convergence

## Example

Determine the radius and interval of convergence of the infinite series

## Complete Solution

### Step 1: Apply Ratio Test

The ratio test gives us:

The ratio test tells us that the power series converges only when

or . Therefore, the **radius of convergence** is 4.

### Step 2: Test End Points of Interval to Find Interval of Convergence

The inequality can be written as -7 < *x* < 1. By the ratio test, we know that the series converges on this interval, but we don't know what happens at the points *x* = -7 and *x* = 1.

At *x* = -7, we have the infinite series

This series diverges by the test for divergence.

At *x* = 1, we have the infinite series

This series also diverges by the test for divergence.

Therefore, the interval of convergence is -7 < *x* < 1.

## Explanation of Each Step

## Possible Challenges

### What Convergence Test Should Be Used?

For most problems, the ratio test can be used initially. If the ratio test yields an *interval* for the domain, we need to use other convergence tests to explore what the domain could be at the end points of the interval.

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