# Calculus 2 : Alternating Series

## Example Questions

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### Example Question #1 : Alternating Series

By definition, an Alternating Series is a series of the form-

Explanation:

This type of series we can frequently check for convergence/divergence using the Alternating Series Test.

The terms with an odd value for  become negative since  and the terms with an even value for  are positive. This creates the alternating signs to occur within the sum.

### Example Question #2 : Alternating Series

Differentiate the following function.

Explanation:

To differentiate the function we will need to use the Power Rule which states:

Looking at our function we can first simplify the equation.

Applying the Power Rule we get:

### Example Question #4 : P Series

Does the series  converge conditionally, absolutely, or diverge?

Converge Absolutely.

Diverges.

Cannot tell with the given information.

Converge Conditionally.

Does not exist.

Converge Conditionally.

Explanation:

The series converges conditionally.

The absolute values of the series  is a divergent p-series with .

However, the the limit of the sequence  and it is a decreasing sequence.

Therefore, by the alternating series test, the series converges conditionally.

### Example Question #3 : Alternating Series

Find the interval of convergence of  for the series .

Explanation:

Using the root test,

Because 0 is always less than 1, the root test shows that the series converges for any value of x.

Therefore, the interval of convergence is:

### Example Question #3 : Alternating Series

Determine whether

converges or diverges, and explain why.

Divergent, by the test for divergence.

Divergent, by the comparison test.

More tests are needed.

Convergent, by the alternating series test.

Convergent, by the -series test.

Convergent, by the alternating series test.

Explanation:

We can use the alternating series test to show that

converges.

We must have   for  in order to use this test. This is easy to see because  is in for all  (the values of this sequence are ), and sine is always nonzero whenever sine's argument is in .

Now we must show that

1.

2.  is a decreasing sequence.

The limit

implies that

so the first condition is satisfied.

We can show that  is decreasing by taking its derivative and showing that it is less than  for :

The derivative is less than , because  is always less than , and that  is positive for , using a similar argument we used to prove that  for . Since the derivative is less than  is a decreasing sequence. Now we have shown that the two conditions are satisfied, so we have proven that

converges, by the alternating series test.

### Example Question #4 : Alternating Series

For the series:  , determine if the series converge or diverge.  If it diverges, choose the best reason.

Explanation:

The series given is an alternating series.

Write the three rules that are used to satisfy convergence in an alternating series test.

For :

The first and second conditions are satisfied since the terms are positive and are decreasing after each term.

However, the third condition is not valid since  and instead approaches infinity.

### Example Question #4 : Alternating Series

Write a series expression for  terms of the following sequence.

This sequence can't be represented as a series.

Explanation:

If we look at this sequence

The first thing we should notice is that it is alternating from positive to negative. This means that we will have

.

The second thing we should notice is that the sequence is increasing in powers of 2.

Thus we will also have

.

Now we can combine these statements and write them in terms of a series.

We can now simplify this into

.

### Example Question #5 : Alternating Series

Determine whether the series is convergent or divergent:

The series is divergent.

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

The series is (absolutely) convergent.

The series is conditionally convergent.

The series is divergent.

Explanation:

To determine whether this alternating series converges or diverges, we must use the Alternating Series test, which states that for the series

and ,

where  for all n, if  and  is a decreasing sequence, then the series is convergent.

First, we must identify , which is . When we take the limit of  as n approaches infinity, we get

Notice that for the limit, the negative power terms go to zero, so we are left with something that does not equal zero.

Thus, the series is divergent because the test fails.

### Example Question #72 : Polynomial Approximations And Series

Determine whether the series converges or diverges:

The series is conditionally convergent.

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

The series is divergent.

The series is (absolutely) convergent.

The series is divergent.

Explanation:

To determine whether the series converges or diverges, we must use the Alternating Series test, which states that for

- and  where  for all n - to converge,

must equal zero and  must be a decreasing series.

For our series,

because it behaves like

.

The test fails because  so we do not need to check the second condition of the test.

The series is divergent.

### Example Question #6 : Alternating Series

Determine if the following series is convergent or divergent:

Divergent according to the alternating series test

Inconclusive according to the alternating series test

Divergent according to the ratio test

Convergent according to the alternating series test

Convergent according to the alternating series test

Explanation:

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1. {} is a decreasing sequence, or in other words

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

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