# Convergence and Divergence

Convergence and Divergence – Introduction to Series
Convergence and Divergence – Introduction to Series

## Example Questions

### Example Question #61 : Ratio Test

Use the ratio test to find out if the following series is convergent:

Determine the convergence of the series based on the limits.

Solution:

1. Ignore constants and simplify the equation (canceling out what you can).

2. Once the equation is simplified, take .

### Example Question #62 : Ratio Test

Use the ratio test to find out if the following series is convergent:

Determine the convergence of the series based on the limits.

Solution:

1. Ignore constants and simplify the equation (canceling out what you can).

2. Once the equation is simplified, take .

### Example Question #63 : Ratio Test

Use the ratio test to find out if the following series is convergent:

Determine the convergence of the series based on the limits.

Solution:

1. Ignore constants and simplify the equation (canceling out what you can).

2. Once the equation is simplified, take .

### Example Question #64 : Ratio Test

Use the ratio test to find out if the following series is convergent:

Determine the convergence of the series based on the limits.

Solution:

1. Ignore constants and simplify the equation (canceling out what you can).

2. Once the equation is simplified, take .

### Example Question #65 : Ratio Test

Use the ratio test to find out if the following series is convergent:

Determine the convergence of the series based on the limits.

Solution:

1. Ignore constants and simplify the equation (canceling out what you can).

2. Once the equation is simplified, take .

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

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

Divergent

Inconclusive

Convergent

Neither

Both

Convergent

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

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

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 #66 : Ratio Test

Determine if the series converges or diverges:

Neither converges nor diverges.

Converges

Conditionally converges.

Diverges

There is not enough information to determine convergency.

Converges

The ratio test states that if you take the n+1 term of the series and divide it by the n term, and then take the limit as n approaches infinity and if you take the absolute value of your answer and if it less than 1, it converges.

If it is greater than 1, it diverges.

If it is 1, the test is inconclusive.

The n+1 term is .

Note that you are substituting n+1 for n and so you will distribute the term by 2 and 3 respectively. Dividing the n+1 term by the n term gives you the following: , which when multiplied out gives us the following:

.

To use the ratio test, we must take the limit of this term as n approaches infinity. From inspection, we can see that the denominator is increasing much faster than the numerator (there are more n terms) and so the limit as n appraoches infinity is 0. Since the absolute value of 0 is less than 1, the series converges.

### Example Question #67 : Ratio Test

Determine what the limit is using the Ratio Test.

To determine what this series converges to

we need to remember the ratio test.

The ratio test is as follows.

Suppose we a series . Then we define,

.

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.

.

Thus the limit of this series is using the ratio test.

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