# 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

Next: Videos on The Ratio Test