Alternating Series Test
So far, we've only looked at positive terms. What if the signs flip-flop?
What is Alternating?
An alternating series has terms that switch signs every time. Usually, this comes from a or term.
We separate the sign from the magnitude. We define as the positive part.
So for , the part we test is just .
The Alternating Series Test
Because the terms cancel each other out, it's easier for an alternating series to converge than a positive one.
Convergence Conditions
The series converges if:
- 1Limit is Zero
- 2Decreasing(eventually)
A Critical Warning
AST Does NOT Prove Divergence
The Alternating Series Test only tells you if something Converges. If the conditions fail, the test is inconclusive... usually.
If Condition 1 Fails ():
The series Diverges, but not because of AST. It diverges by the Divergence Test.
Worked Examples
Example 1: Alternating Harmonic Series
Check .
Identify .
Both conditions met. Series Converges.
Example 2: The "Fake" Alternating Series
Check .
Identify .
Limit is not zero. Series Diverges (by Divergence Test).
Example 3: Using Derivatives
Check .
It's hard to see if . Let's use a derivative on .
Since is negative for , the function is decreasing.
Decreasing + Limit is 0 = Converges.
Practice Quiz
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