First, find the limit: Which of the following is true?
for for neither
Describe the behavior of the series
Converges by AST Diverges by Test for
Divergence No Conclusion from AST
We use the Alternating Series Test to determine convergence of infinite series.
An alternating series is an infinite series whose terms alternate signs. A typical alternating series has the form where for all . We will refer to the factor as the alternating symbol.
Some examples of alternating series are
- This is the alternating harmonic series.
- This is an alternating geometric series with . Since , we know from our earlier study of geometric series that this alternating series converges and the sum is
- In this example, could be replaced by the standard alternating symbol .
- This is an alternating -series.
The Alternating Series Test
Alternating Series Test (AST)
The alternating series, converges if the following two conditions are met:
The alternating series, converges if the following two conditions are met:
- ,
- the terms, , are decreasing, i.e., for all .
If the first condition does not hold, i.e., if , then the alternating series diverges (by
the Test for Divergence).
The second condition can be weakened to say the terms, , are eventually decreasing,
i.e., decreasing from some term onward.
If the first condition is satisfied but the second condition is not, then the AST gives
no conclusion about the alternating series.
example 1 Determine if alternating harmonic series converges or diverges:
First, so the terms go to zero.
Second, for all , so the terms are decreasing.
Since both conditions are satisfied, the alternating harmonic series converges by the AST. Moreover, it is known that the alternating harmonic series converges to the value , as we will see in the section on Power Series.
First, so the terms go to zero.
Second, for all , so the terms are decreasing.
Since both conditions are satisfied, the alternating harmonic series converges by the AST. Moreover, it is known that the alternating harmonic series converges to the value , as we will see in the section on Power Series.
(problem 1a) Determine whether the series converges or diverges:
First, find the limit: Which of the following is true?
First, find the limit: Which of the following is true?
(problem 1b) Determine whether the series converges or diverges:
First, find the limit: Which of the following is true?
First, find the limit: Which of the following is true?
for for neither
Describe the behavior of the series
Converges by AST Diverges by Test for
Divergence No Conclusion from AST
(problem 1c) Determine whether the series converges or diverges:
First, find the limit: Which of the following is true?
First, find the limit: Which of the following is true?
for for neither
Describe the behavior of the series
Converges by AST Diverges by Test for
Divergence No Conclusion from AST
(problem 1d) Determine whether the series converges or diverges:
First, find the limit: Which of the following is true?
First, find the limit: Which of the following is true?
for for neither
Describe the behavior of the series
Converges by AST Diverges by Test for
Divergence No Conclusion from AST
example 2 Let . Determine whether the alternating -series converges or diverges:
First, so the terms approach zero.
Second,
First, so the terms approach zero.
Second,
so the terms are decreasing for all .
Since both conditions are satisfied, the alternating -series converges by the
AST.
Absolute Versus Conditional Convergence
In general, the presence of the alternating symbol, , helps a series to converge. The alternating harmonic series and its non-alternating counterpart, the harmonic series, provide the quintessential example of this. The harmonic series diverges, but with the addition of the alternating symbol, the alternating harmonic series converges. This leads us to investigate alternating series further by asking the question “does a convergent alternating series require the alternating symbol to converge?” If an alternating series converges without the alternating symbol, then we say that it is absolutely convergent. On the other hand, if a convergent alternating series diverges without the alternating symbol (like the harmonic series), then we say that the alternating series is conditionally convergent. We formalize these terms in the definition below.
Absolute vs Conditional Convergence Consider the alternating series where for all
.
- If the non-alternating counterpart, , converges, then we say that the original alternating series, , converges absolutely.
- If the non-alternating counterpart, , diverges, then we say that the original alternating series, , converges conditionally.
example 3 Determine whether the alternating -series converges absolutely or
conditionally: where .
In our study of non-alternating -series in the section on the Integral Test, we discovered that the standard -series, , converges if and diverges if . In the previous example, we showed that alternating -series converge for all . Hence, we can say that if , the alternating -series converges absolutely and if , the alternating -series converges conditionally. In particular, the alternating harmonic series () converges conditionally.
In our study of non-alternating -series in the section on the Integral Test, we discovered that the standard -series, , converges if and diverges if . In the previous example, we showed that alternating -series converge for all . Hence, we can say that if , the alternating -series converges absolutely and if , the alternating -series converges conditionally. In particular, the alternating harmonic series () converges conditionally.
example 4 Determine whether the alternating series converges absolutely, converges
conditionally or diverges.
First, we consider the non-alternating counterpart of the given series: This series diverges by the Limit Comparison Test with the divergent harmonic series since,
First, we consider the non-alternating counterpart of the given series: This series diverges by the Limit Comparison Test with the divergent harmonic series since,
by the ratio of the leading coefficients. Since , the non-alternating counterpart also diverges by the LCT. Next, we check for conditional convergence using the Alternating Series Test. The first condition is satisfied since For the second condition, we must determine if the terms are (eventually) decreasing. To do this, we will compute the derivative (using the quotient rule) and check to see if it is negative:
The derivative is negative for since the numerator is negative and the denominator is positive when . Hence, the terms of the series are eventually decreasing, and the second condition is also satisfied. Finally, since both conditions of the AST are satisfied, we can conclude that the original alternating series converges conditionally.(problem 4) Determine whether the following series converges absolutely, converges
conditionally or diverges: .
Does its non-alternating counterpart converge?
Does its non-alternating counterpart converge?
Yes No
Why?
Limit Comparison Test with the harmonic series -series with geometric
series with
Does the original alternating series converge absolutely?
Yes No
We now use the AST to check for conditional convergence.
First, find the limit:
The derivative of is
The derivative is negative for .
Are the terms eventually decreasing?
Yes No
Does the original alternating series converge conditionally?
Yes No
Video Lesson
Here is a detailed, lecture style video on the AST:
_
More Problems
(problem 5) Determine whether the following series converges absolutely, converges
conditionally or diverges: .
Does its non-alternating counterpart converge?
Does its non-alternating counterpart converge?
Yes No
Why?
-series with -series with geometric series with
Does the original alternating series converge absolutely?
Yes No
(problem 6) Determine whether the following series converges absolutely, converges
conditionally or diverges: .
Does its non-alternating counterpart converge?
Does its non-alternating counterpart converge?
Yes No
Why?
-series with -series with
Does the original alternating series converge absolutely?
Yes No
We now use the AST to check for conditional convergence.
First, find the limit: Which of the following is true?
for for neither
Does the series converge conditionally?
Yes No
(problem 7) Determine whether the following series converges absolutely, converges
conditionally or diverges: .
Does its non-alternating counterpart converge?
Does its non-alternating counterpart converge?
Yes No
Why?
Limit Comparison Test with the harmonic series -series with geometric
series with
Does the original alternating series converge absolutely?
Yes No
We now use the AST to check for conditional convergence.
First, find the limit: Which of the following is true?
for for neither
Does the series converge conditionally?
Yes No
(problem 8) Determine whether the following series converges absolutely, converges
conditionally or diverges: .
Does its non-alternating counterpart converge?
Does its non-alternating counterpart converge?
Yes No
Why?
Direct Comparison Test with -series -series with geometric series with
Does the original alternating series converge absolutely?
Yes No
(problem 9) Determine whether the following series converges absolutely, converges
conditionally or diverges: .
Does its non-alternating counterpart converge?
Does its non-alternating counterpart converge?
Yes No
Why?
Limit Comparison Test with -series -series with geometric series with
Does the original alternating series converge absolutely?
Yes No
We now use the AST to check for conditional convergence.
First, find the limit: Which of the following is true?
for for neither
Does the series converge conditionally?
Yes No
(problem 10) Determine whether the following series converges absolutely, converges
conditionally or diverges: .
Does its non-alternating counterpart converge?
Does its non-alternating counterpart converge?
Yes No
Why?
Limit Comparison Test with -series -series with Test for Divergence
Does the original alternating series converge absolutely?
Yes No
Find the limit:
Do we need to check the second condition of the AST?
Yes No
What can we say about the series ?
Converges Conditionally Diverges No
Conclusion