# A Simple Ratio Test Example

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## Question

Using only the ratio test, determine whether or not the recursive sequence



converges or diverges.

## Complete Solution

Applying the ratio test yields



But



Therefore,



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 can use



and substitute this into our limit.

### Step (2)

In Step (2), we only cancel the  in the numerator and denominator.

### Step (3)

First observe that



Dividing everything by the square root of  we obtain



### Step (4)

In Step (4) we only evaluate the limit:



which equals zero because the numerator is a constant and the denominator goes to infinity.

### Step (5)

In Step (5) we apply the Squeeze Theorem.

## Potential Challenge Areas

### Getting Started

Because the question asks us to apply the ratio test, we know that we will start our solution by using the formula



### Recursive Formula

Most problems involving convergence tests don't involve recursive formulas. But with the ratio test, we apply



and use the given recursion equation for . In our case, our recursion equation is



which we substitute into the numerator, allowing us to cancel the  in the numerator and denominator. This trick is a bit harder to apply for the other convergence tests.

Previous: The Ratio Test Flowchart

Next: Ratio Test Example with an Exponent