# Calculus 2 : Types of Series

## Example Questions

### Example Question #2 : Harmonic Series

Does the following series converge?

Cannot be determined

No

Yes

Yes

Explanation:

The series converges. The given problem is the alternating harmonic series, which converges by the alternating series test.

### Example Question #3 : Harmonic Series

Which of the following tests can be used to (successfully) test for the convergence/divergence of the harmonic series?

The Root Test

The Limit Test for Divergence

None of the given tests can be used.

The Ratio Test

The Integral Test

The Integral Test

Explanation:

Only the Integral Test will work on the Harmonic Series, .

To use the Integral Test, we evaluate

, which shows that the series diverges.

Since , the Limit Test for Divergence fails.

The Ratio Test and the Root Test will always yield the same conclusion, so if one test fails, the both fail and vise versa.

For the Ratio Test,

. Since the result of the limit is , both tests fail.

### Example Question #4 : Harmonic Series

Let's say you are given harmonic series in the following form:

;

You are then asked to determine if the series converges, or diverges. For what values of p would this series be convergent? Assume p>0.

Explanation:

The given series is called generalized harmonic series.

The series converges, if , and diverges, if .

### Example Question #3001 : Calculus Ii

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 #1 : 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?

Cannot tell with the given information.

Converge Absolutely.

Does not exist.

Converge Conditionally.

Diverges.

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 #1 : 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 #3004 : Calculus Ii

Determine whether

converges or diverges, and explain why.

Divergent, by the test for divergence.

Divergent, by the comparison test.

Convergent, by the -series test.

More tests are needed.

Convergent, by the alternating 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 #3005 : Calculus Ii

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 #1 : 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

.