# Calculus 2 : Types of Series

## Example Questions

1 2 3 4 5 7 Next →

### Example Question #231 : Series In Calculus

Determine how many terms need to be added to approximate the following series within

Explanation:

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that

Step 1:

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

Solution:

2. {} is a decreasing sequence

Since the factorial is always increasing, the sequence is always decreasing.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until

so 7 terms are needed to approximate the sum within .001.

### Example Question #6 : Sequences & Series

Determine how many terms need to be added to approximate the following series within

Explanation:

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that

Step 1:

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

Solution:

1.

2. {} is a decreasing functon, since a factorial never decreases.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until

4 needs to be added to approximate the sum within .

### Example Question #241 : Series In Calculus

Determine how many terms need to be added to approximate the following series within

Explanation:

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that

Step 1:

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

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until

7 terms are needed to approximate the sum within .001

### Example Question #242 : Series In Calculus

Determine how many terms need to be added to approximate the following series within

Explanation:

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that

Step 1:

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

Solution:

1.

2. {{b_{n}} is a decreasing sequence. A factorial always increases as n increases, so each term will decrease as n increases.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until

4 terms need to be added to approximate the sum within .001

### Example Question #243 : Series In Calculus

Determine whether the series converges or diverges:

The series is conditionally convergent

The series is divergent

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

The series is (absolutely) convergent

The series is (absolutely) convergent

Explanation:

To determine whether the given alternating series converges or diverges, we must perform the Alternating Series test, which states that for a given series

and  or , for the series to converge,  and  must be decreasing.

To start, we must take the limit of  as n approaches infinity:

because (the numerator goes to zero).

Next, we must see if  is decreasing. Simply increasing  to  does not clearly show whether the function is decreasing because both the numerator and denominator increase. So, we must find the first derivative and see if it is negative:

,

and was found using the following rules:

,

The derivative is always negative from  to , so the sequence   is decreasing. The series is (absolutely) convergent because it passed both parts of the test.

### Example Question #244 : Series In Calculus

Determine whether the series is convergent or divergent:

The series is divergent

The series may be conditionally convergent, (absolutely) convergent, or divergent

The series is (absolutely) convergent

The series is conditionally convergent

The series is divergent

Explanation:

To determine whether the series is convergent or divergent, we must use the Alternating Series test, which states that for a given series , where  or , if  and  is decreasing, then the series is convergent.

First, we must evaluate the limit of  as  approaches infinity:

Therefore, the test fails and series is divergent.

### Example Question #245 : Series In Calculus

Determine whether the following series is convergent or divergent:

The series is conditionally convergent

The series is (absolutely) convergent

The series is divergent

The series may be (absolutely) convergent, conditionally convergent, or divergent

The series is divergent

Explanation:

To determine whether the series is convergent or divergent, we must use the Alternating Series test, which states that for a given series , where  or , if  and  is decreasing, then the series is convergent.

First, we must evaluate the limit of  as  approaches infinity:

The test fails and the series is therefore divergent.

### Example Question #246 : Series In Calculus

Determine whether the series is convergent or divergent:

The series is conditionally convergent

The series is divergent

The series is (absolutely) convergent

The series may be (absolutely) convergent, conditionally convergent, or divergent