convergence/divergence tests

Nth Term Test for Divergence

• if {aₙ} does not converge to 0 (take the limit), then the series diverges

• if {aₙ} converges to 0, the Nth Term Test is inconclusive

Harmonic Series

1/n is convergent

Geometric Series Test

• if | r | ≥ 1, the series diverges

• if 0 < | r | < 1, the series converges to a/1-r

Integral Test

PRE−REQS:

• f must be positive, continuous, decreasing for x ≥ 1

• ∫f(x) dx converges (bound is 1 to ∞) → the series converges

• ∫f(x) dx diverges (bound is 1 to ∞) → the series diverges

p-Series

• converges if p > 1

• diverges if 0 < p ≤ 1

Direct Comparison Test

let 0 < aₙ < bₙ for all n:

• if bₙ converges, then aₙ converges

• if aₙ diverges, then bₙ diverges

Limit Comparison Test

• let aₙ > 0 and bₙ > 0

• L (the limit of aₙ/bₙ) must be a finite and positive value

Alternating Series Test

• series must be alternating: (−1)ⁿ+¹ aₙ or (−1)ⁿ aₙ

• if the limit of the series is 0 and a_n+1 ≤ aₙ for all n, the series converges

Root Test