# AP Calculus BC : Ratio Test and Comparing Series

## Example Questions

### Example Question #1 : Ratio Test And Comparing Series

Determine if the following series is divergent, convergent or neither.

Divergent

Inconclusive

Neither

Convergent

Both

Convergent

Explanation:

In order to figure out if

is divergent, convergent or neither, we need to use the ratio test.

Remember that the ratio test is as follows.

Suppose we have a series . We define,

Then if

, the series is absolutely convergent.

, the series is divergent.

, the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply the ratio test to our problem.

Let

and

Now

Now lets simplify this expression to

.

Since

.

We have sufficient evidence to conclude that the series is convergent.

### Example Question #2 : Ratio Test And Comparing Series

Determine if the following series is divergent, convergent or neither.

Divergent

Convergent

Inconclusive

Both

Neither

Divergent

Explanation:

In order to figure if

is convergent, divergent or neither, we need to use the ratio test.

Remember that the ratio test is as follows.

Suppose we have a series . We define,

Then if

, the series is absolutely convergent.

, the series is divergent.

, the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply the ratio test to our problem.

Let

and

Now

.

Now lets simplify this expression to

.

Since ,

we have sufficient evidence to conclude that the series is divergent.

### Example Question #3 : Ratio Test And Comparing Series

Determine if the following series is divergent, convergent or neither.

Both

Neither

Inconclusive

Convergent

Divergent

Divergent

Explanation:

In order to figure if

is convergent, divergent or neither, we need to use the ratio test.

Remember that the ratio test is as follows.

Suppose we have a series . We define,

Then if

, the series is absolutely convergent.

, the series is divergent.

, the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply the ratio test to our problem.

Let

and

.

Now

.

Now lets simplify this expression to

.

Since ,

we have sufficient evidence to conclude that the series is divergent.

### Example Question #4 : Ratio Test And Comparing Series

Determine if the following series is convergent, divergent or neither.

More tests are needed.

Convergent

Inconclusive

Divergent

Neither

Divergent

Explanation:

To determine if

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

the series is absolutely convergent (and therefore convergent).

the series is divergent.

the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

Now lets simplify this.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series diverges.

### Example Question #5 : Ratio Test And Comparing Series

Determine if the following series is divergent, convergent or neither.

Divergent

Inconclusive

Neither

More tests are needed.

Convegent

Convegent

Explanation:

To determine if

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

the series is absolutely convergent (and thus convergent).

the series is divergent.

the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

.

Now lets simplify this.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series converges.

### Example Question #71 : Convergence And Divergence

Determine if the following series is convergent, divergent or neither.

Inconclusive

More tests needed.

Convergent

Divergent

Neither

Divergent

Explanation:

To determine if

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

the series is absolutely convergent (therefore convergent).

the series is divergent.

the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

Now lets simplify this.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series diverges.

### Example Question #7 : Ratio Test And Comparing Series

Determine if the following series is divergent, convergent or neither.

Inconclusive

Convergent

More tests are needed.

Divergent

Neither

Divergent

Explanation:

To determine if

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

the series is absolutely convergent (and thus convergent).

the series is divergent.

the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can simplify the expression to be

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series diverges.

### Example Question #8 : Ratio Test And Comparing Series

Determine of the following series is convergent, divergent or neither.

Inconclusive.

More tests are needed.

Neither

Divergent

Convergent

Divergent

Explanation:

To determine whether this series is convergent, divergent or neither

we need to remember the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

the series is absolutely convergent (and therefore convergent).

the series is divergent.

the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

Now lets simplify this to.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series is divergent.

### Example Question #9 : Ratio Test And Comparing Series

Determine what the following series converges to using the ratio test and whether the series is convergent, divergent or neither.

, and neither.

, and convergent.

, and divergent.

, and convergent.

, and neither.

, and convergent.

Explanation:

To determine whether this series is convergent, divergent or neither

we need to remember the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

the series is absolutely convergent (thus convergent).

the series is divergent.

the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

Now lets simplify this to.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series is convergent.

### Example Question #1 : Ratio Test And Comparing Series

Determine the convergence or divergence of the following series:

The series may be divergent, conditionally convergent, or absolutely convergent.

The series is conditionally convergent.

The series is divergent.

The series (absolutely) convergent.

The series (absolutely) convergent.

Explanation:

To determine the convergence or divergence of this series, we use the Ratio Test:

If , then the series is absolutely convergent (convergent)

If , then the series is divergent

If , the series may be divergent, conditionally convergent, or absolutely convergent

So, we evaluate the limit according to the formula above:

which simplified becomes

Further simplification results in

Therefore, the series is absolutely convergent.