Calculus: Absolute and Conditional Convergence for
Alternating Series,
Absolute Convergence
P
A series an is said to be absolutely convergent if:
X
|an |
converges.
Key idea: If the series of absolute values converges, then the original series also con-
verges.
Conditional Convergence
P
A series an is conditionally convergent if:
•
P
an converges, but
•
P
|an | diverges.
Example: The alternating harmonic series
∞
X (−1)n+1
n=1
n
converges conditionally, because:
• The alternating series test shows convergence.
• The harmonic series
P1
n
diverges, so absolute convergence fails.
Formal Definitions
P P
1. an is absolutely convergent if|an | converges.
P P P
2. an is conditionally convergent if an converges but |an | diverges.
1
Alternating Series,
Absolute Convergence
P
A series an is said to be absolutely convergent if:
X
|an |
converges.
Key idea: If the series of absolute values converges, then the original series also con-
verges.
Conditional Convergence
P
A series an is conditionally convergent if:
•
P
an converges, but
•
P
|an | diverges.
Example: The alternating harmonic series
∞
X (−1)n+1
n=1
n
converges conditionally, because:
• The alternating series test shows convergence.
• The harmonic series
P1
n
diverges, so absolute convergence fails.
Formal Definitions
P P
1. an is absolutely convergent if|an | converges.
P P P
2. an is conditionally convergent if an converges but |an | diverges.
1
