Absolute Convergence

Summary

A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges. Absolute convergence implies ordinary convergence. Series that converge but not absolutely are called conditionally convergent.

Prerequisites

Infinite Series, Alternating Series, Ratio Test

Definition

• Absolute convergence: $\sum |a_n|$ converges.
• Conditional convergence: $\sum a_n$ converges but $\sum |a_n|$ diverges.

Theorem

If $\sum |a_n|$ converges, then $\sum a_n$ converges. The converse is false: the alternating harmonic series converges, but $\sum 1/n$ diverges.

Absolutely convergent series may be rearranged freely without changing the sum; conditionally convergent series may not (Riemann rearrangement theorem).

Worked Example

$\sum (1/2{)}^n$ converges absolutely.

$\sum (-1{)}^{n+1}/n$ converges conditionally.

$\sum x^n/n!$ converges absolutely for every real $x$ (ratio test / exponential series).

Common Mistakes

• Equating “converges” with “converges absolutely.”
• Rearranging conditionally convergent series and expecting the same sum.

Connections

• Tests that imply absolute convergence: Ratio Test, Root Test, comparison on $|a_n|$
• Alternating Series for conditional examples

References

Absolute vs conditional convergence is in OpenStax Calculus Volume 2.^[1]