convergence/divergence tests

geometric series test (GT)

• converges if |r| < 1

• diverges if |r| ≥ 1

divergence test (NT)

• diverges if lim ≠ 0

• inconclusive if lim = 0

integral test (IT)

• 1-1 ratio

• both converge or diverge

p-series test (PT)

• converges if p > 1

• diverges if p ≤ 1

direct comparison test (DCT)

• smaller diverges = whole thing diverges

• larger converges = whole thing converges

• larger diverges/smaller converges = inconclusive

limit comparison test (LCT)

• compare an to bn

• ONLY VALID if lim an/bn is a positive, finite number

ratio test (RT)

• if lim | a*r^(n+1) / a*r^n |:

  • lim |r|

    • < 1 → converges absolutely;

    • > 1 or =infinity → diverges;

    • = 1 → inconclusive.

root test (NRT)

• lim of nth root of |an|:

  • < 1 → converges absolutely

  • > 1 or =infinity → diverges

  • =1 → inconclusive

absolute convergence test (ACT)

• Σ |an| converges → original series an converges

• Σ |an| diverges → test inconclusive

alternating series test (AST)

• n+1 / n —> goal is to cancel out!

• if lim bn =0 → converges

• if lim bn ≠0 → AST inconclusive