AP Calculus BC - Sequences & Series Convergence Test Flashcards

Memorization practice for criteria for convergence/divergence

Nth Term Test for Divergence

lim n→∞ a_n ≠ 0 ; ∑a_n diverges

lim n→∞ a_n = 0 ; inconclusive, try another test

p-series

(a_n = 1/n^p, n ≥ 1)

p > 1; converges

p ≤ 1; diverges

Geometric Series

(a_n = ar^n-1, n ≥ 1)

|r| < 1 ; converges to a_n = ar^k / 1 - r

|r| ≥ 1 ; diverges

Alternating Series

(a_n = (-1)ⁿ b_n or a_n = (-1)ⁿ⁺¹ b_n, b_n ≥ 0)

lim n→∞ b_n = 0 AND b_n+1/b_n < 1 ; converges

Comparison Test

Pick b_n to compare to a_n

∑b_n converges; 0 ≤ a_n ≤ b_n ; converges

∑b_n converges; NOT 0 ≤ a_n ≤ b_n ; inconclusive, pick a better comparison

∑b_n diverges; 0 ≤ b_n ≤ a_n ; diverges

Limit Comparison Test

lim n→∞ a_n/b_n = L > 0, ∑b_n converges; converges

∑b_n diverges; diverges

Integral Test

a_n = f(n)

Must be continuous, positive, and decreasing

∫_a^∞ f(n)dn converges; ∑a_n converges

∫_a^∞ f(n)dn diverges; ∑a_n diverges

Ratio Test

lim n→∞ |a_n+1/a_n| ≠ 1

lim n→∞ |a_n+1/a_n| < 1 ; absolutely converges

lim n→∞ |a_n+1/a_n| > 1 ; diverges