# Ratio Test Example with an Exponent

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## Question

Using only the ratio test, determine whether or not the series converges, diverges, or yields no conclusion.

## Complete Solution

Applying the ratio test yields Since the limit equals , the ratio test tells us that the series converges.

## Explanation of Each Step

### Step (1)

To apply the ratio test, we must evaluate the limit In our problem, we have and we substitute them into our limit.

### Step (2)

In Step (2), we use a little algebraic manipulation to make things easier to look at ### Step (3)

Step (3) uses a property of absolute values. Recall that for real numbers and , ### Step (4)

Step (4) uses a property of limits values. Recall that for functions and , that ### Step (5)

Here we evaluate a limit: ### Step (6)

Some algebraic manipulation helps us see how we can simplify our problem. Recall that, using laws of exponentials, that for real numbers , that so Previous: A Simple Ratio Test Example

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