AP CALC BC (Infinite Series Tests)

nth Term Test

limit does not approach 0 (diverges)

Geometric Series

• converges if |r|<1 to a/1-r

• diverges if |r|>1 or equal to 1

P-Series

• converges if p>1

• diverges if 0<p<1 or equal to 1 (Harmonic)

Alternating Series Test

Converges if the limit of the terms approaches zero and the absolute value of the terms is decreasing.

Comparison Test

• if converging OG is smaller and if diverging OG is larger.

Limit Comparison Test

• 0<L<infinity = both series converge or both series diverge

• zero + comparison converges = OG converges

• infinity + comparison diverges =OG diverges

Ratio Test

• L<1 then convergent

• L>1 or infinity then divergent

• L=1 is inconclusive

Root Test

• L<1 then convergent

• L>1 or infinity then divergent

• L=1 is inconclusive

Integral Test

• positive (>0), continuous, and decreasing (the next term is smaller than or equal to it)

• series converges if integral converges

• series diverges if integral diverges

*** sum of series does NOT equal what the integral converges to ***

Infinite Sequences

• converges if limit exists as a finite number

Alternating Sequence

• only converge if limit = 0