Absolute Convergence
A stronger form of convergence where the signs of the terms don't matter.
Introduction
Some series converge only because their terms cancel each other out (like the Alternating Harmonic Series). Others converge simply because their terms get small fast enough, regardless of sign. This distinction is between Conditional and Absolute convergence.
Interactive: Partial Sums
Green: (Converges fast). Blue: (Converges slowly, conditionally). Red: (Diverges).
Definitions & Facts
Absolute Convergence
A series is absolutely convergent if the series of absolute values converges.
Conditional Convergence
A series is conditionally convergent if converges but diverges.
Example: Alternating Harmonic Series.
Why does it matter?
- Rearrangement: Absolutely convergent series can be rearranged without changing the sum. Conditionally convergent series cannot—rearranging terms can change the sum to any number!
- Tests: Ratio and Root tests only test for absolute convergence.
Worked Examples
Example 1: Checking Types (Level 1)
Determine if is absolutely convergent, conditionally convergent, or divergent.
Step 1: Check Absolute
Take absolute values: .
Step 2: Test Convergence
This is a p-series with .
Therefore, converges.
Result: Absolutely Convergent.
(This implies the original series also converges).
Example 2: Conditional Convergence (Level 2)
Classify .
Step 1: Check Absolute
.
This is the Harmonic Series, which diverges.
So, it is NOT absolutely convergent.
Step 2: Check Original
Use Alternating Series Test on .
is decreasing and .
So, it converges.
Result: Conditionally Convergent.
Example 3: Tricky Case (Level 3)
Classify .
Step 1: Check Absolute
.
Step 2: Comparison
We know .
So .
Since converges (p=2), by Comparison Test, the absolute series converges.
Result: Absolutely Convergent.
Note: We didn't need to check "conditional" because absolute implies convergence.
Practice Quiz
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