math series flashcards

what to look for in alternating series

(-1)ⁿ⁻¹ or (-1)ⁿ

when does an alt series converge/diverge

converges when lim n→∞ =0 & aₙ₊₁ ≤ aₙ

use nth term to show divergence when lim n→∞ ≠ 0

what to look for to apply ratio test

factorials & exponentials

when does ratio test converge/diverge

converges when lim n→∞ |aₙ₊₁ / aₙ| < 1

diverges when lim n→∞ |aₙ₊₁ / aₙ| > 1

(is inconclusive if lim = 1)

what to look for to apply root test

∑(aₙ)ⁿ

when does root test converge/diverge

converges when lim n→∞ ⁿ√|aₙ| < 1

diverges when lim n→∞ ⁿ√|aₙ| > 1

(is inconclusive if lim = 1)

what to look for in integral test

the series is integrable (or nothing else works)

what are the conditions to apply integral test

aₙ is positive, dec, and continuous

when does integral test converge/diverge

converges if lim b→∞ ∫ (from 1 to b) aₙ = #

diverges if lim b→∞ ∫ (from 1 to b) aₙ = ∞

what to look for in p-series

∑ 1/n^p

when do p series (1/n^p) converge/diverge

converges when p < 1

diverges when p ≥ 1

what to look for in direct comparison

aₙ is < or > a similar series

when do direct comparison series converge/diverge

converge if aₙ (original series) < bₙ (comparison series) & we know bₙ converges

diverge if aₙ > bₙ & we know bₙ diverges

what to look for in limit comparison

polynomial/polynomial, sin(1/n), tan(1/n)

when do limit comparison series converge/diverge

if lim n→∞ | aₙ / bₙ | is positive and finite, the series match

if the limit is 0 and your comparison series converges, the original series converges

if the limit is ∞ and the comparison series diverges, the original series diverges

what to look for in geometric series

∑ a(r)ⁿ

when do geometric series converge/diverges

converges when |r| < 1

diverges when |r| ≥ 1

what to look for in a telescoping series

∑aₙ -bₙ

when do telescoping series converge/diverge

converges if the sum = #

diverges if sum = ∞

how do we conclude series tested for divergence

diverges if lim n →∞ = ∞ or ≠ 0

inconclusive if lim n →∞ = 0