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## Example

Determine the values of *r* so that the sequence

is convergent.

## Complete Solution

We can solve this problem by considering cases for the value of *r*.

### Case 1

If *r* > 1, then *r ^{n}* tends to infinity as

*n*tends to infinity. The sequence is

**divergent**in this case.

### Case 2

If *r*= 1, then

so the sequence is **convergent** for this case.

### Case 3

If , then

so the sequence is **convergent** for this case.

### Case 4

If , then

does not exist, so the sequence is **divergent** for this case.

### Case 5

If , then tends to negative infinity as does not tend to a single finite number. The sequence is **divergent** in this case.

### Summary

Therefore, the sequence is convergent when .

## Explanation of Each Step

### Case 1

Consider the case when . Then our sequence becomes

which tends to infinity.

### Case 2

Here we are using a fundamental property of limits, that the limit of a constant equals that constant:

for any constant and .

### Case 3

Consider the case when . Then our sequence becomes

which tends to zero.

Similarly, if . Then our sequence becomes

which also tends to zero.

### Case 4

In this case, we have the sequence

As approaches infinity the sequence does not approach a unique value, so the limit does not exist.

### Case 5

This case is similar to Case 1. Consider the case when . Then our sequence becomes

The terms alternate between positive and negative numbers, and do not tend to a single finite number.

## Possible Challenge Areas

### Connecting Results to Definition of Convergence

In each of the cases, we used a limit to determine whether the sequence is convergent. According to our definition of convergence of a sequence, as long as our respective limits exist, then the sequence converges.

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