Calculus 2 : Alternating Series
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Example Question #21 : Alternating Series
Determine how many terms need to be added to approximate the following series within
This is an alternating series test.
In order to find the terms necessary to approximate the series within 
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.
-
- {
} 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 #22 : Alternating Series
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
To determine whether the given alternating series converges or diverges, we must perform the Alternating Series test, which states that for a given series
To start, we must take the limit of 
Next, we must see if 
and was found using the following rules:
The derivative is always negative from 
Example Question #23 : Alternating Series
Determine whether the series is convergent or divergent:
The series is conditionally convergent
The series is (absolutely) convergent
The series is divergent
The series may be conditionally convergent, (absolutely) convergent, or divergent
The series is divergent
To determine whether the series is convergent or divergent, we must use the Alternating Series test, which states that for a given series 
First, we must evaluate the limit of 
Therefore, the test fails and series is divergent.
Example Question #24 : Alternating Series
Determine whether the following series is convergent or divergent:
The series is (absolutely) convergent
The series may be (absolutely) convergent, conditionally convergent, or divergent
The series is divergent
The series is conditionally convergent
The series is divergent
To determine whether the series is convergent or divergent, we must use the Alternating Series test, which states that for a given series 
First, we must evaluate the limit of 
The test fails and the series is therefore divergent.
Example Question #25 : Alternating Series
Determine whether the series is convergent or divergent:
The series may be (absolutely) convergent, conditionally convergent, or divergent
The series is divergent
The series is (absolutely) convergent
The series is conditionally convergent
The series is divergent
To determine whether the series is convergent or divergent, we must use the Alternating Series test, which states that for a given series 
First, we must evaluate the limit of 
The test fails and therefore the series is divergent.
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