Concept preview

What you will be able to do

1. 01Define absolute and conditional convergence for a series $\sum a_n$: absolute when $\sum|a_n|$ converges, conditional when $\sum a_n$ converges but $\sum|a_n|$ diverges.
2. 02State and use the theorem that absolute convergence implies convergence, and follow a fixed decision procedure that tests the absolute series $\sum|a_n|$ first.
3. 03Classify a given series as absolutely convergent, conditionally convergent, or divergent, avoiding the common error of calling a conditionally convergent series absolutely convergent.

Start with the definitions, stated precisely because the whole lesson turns on them. A series $\sum a_n$ converges ABSOLUTELY if the series of absolute values $\sum|a_n|$ converges. A series $\sum a_n$ converges CONDITIONALLY if $\sum a_n$ itself converges but $\sum|a_n|$ diverges. Notice the logic: conditional convergence requires two facts held at once — the original series converges, and yet its absolute-value companion does not. If neither the series nor its absolute companion converges, the series simply diverges and neither label applies.