Unit 9 Calc BC — Convergence/Divergence Tests

Nth Term Test

Summation of a_n

• Only tests for divergence

• If the limit of a_n as n approaches infinity is not equal to zero; the sequence diverges

Geometric Series

Summation of a* ( r )ⁿ

• |r| < 1 then converges

• |r| >2 then diverges

• The sum (S) = a/(1-r)

Telescoping Series

Summamtion of (a_n - a_n+1)

• Only tests convergence

• Is the limit as n approaches infinity of a_n is equal to some number L, it converges

• Sum (S) = a₁ - L

P-Series Test

Summation of 1/n^p

• if p > 1, it converges

• if 0 < p < 1, it diverges

• if p = 1 it is called a harmonic series, those also diverge

Alternating Series Test (AST)

Summation of (-1)ⁿ * a_n

• Only tests convergence

• If a_n is decreasing and the limit as n approaches infinity is equal to 0, it converges

Integral Test

Summation of a_n where a_n = f(n)

• if the integral from 1 to infinity of f(x) converges, the sum converges

• f the integral from 1 to infinity of f(x) diverges, the sum diverges

Ratio Test

Summation of a_n

• best to use when there is a factorial

• if the limit as n approaches infinity of |(a_n+1) / (a_n)| < 1, it converges

• if the limit as n approaches infinity of |(a_n+1) / (a_n)| > 1 (or infinity), it diverges

• inconclusive if the limit as n approaches infinity of |(a_n+1) / (a_n)| = 1

Root Test

Summation of a_n

• if the limit as n approaches infinity of the nth root of |a_n| < 1, it converges

• if the limit as n approaches infinity of the nth root of |a_n| > 1 (or infinity), it diverges

• inconclusive if the nth root of |a_n| = 1

Direct Comparison Test (DCT)

Summation of a_n

• if 0 < a_n ≤ b_n AND summation of b_n converges, a_n converges

• if 0 < b_n ≤ a_n AND summation of b_n diverges, a_n diverges

Limit Comparison Test

Summation of a_n

• if the limit of a_n / b_n = L > 0 AND the summation of b_n converges, a_n will converge

• if the limit of a_n / b_n = L > 0 AND the summation of b_n diverges, a_n will diverge

• L must be finite and positive

When does a sequence converge

When the limit as n approaches infinity of the sequence equals a value