Series convergence tests sheet
| Test | Converges if | Diverges if | Best used when |
|---|---|---|---|
| Divergence Test | Cannot prove convergence | lim aₙ ≠ 0 | Quick check -if limit ≠ 0, series diverges |
| Integral Test | ∫₁^∞ f(x)dx converges | ∫₁^∞ f(x)dx diverges | f(x) is positive, continuous, decreasing |
| p-Series Test | p > 1 | p ≤ 1 | Series has form Σ 1/nᵖ |
| Direct Comparison | 0 ≤ aₙ ≤ bₙ and Σbₙ converges | aₙ ≥ bₙ ≥ 0 and Σbₙ diverges | Can bound series by known convergent/divergent series |
| Limit Comparison | lim(aₙ/bₙ) = L > 0, finite, and Σbₙ converges | lim(aₙ/bₙ) = L > 0, finite, and Σbₙ diverges | Series "looks like" a known series |
| Alternating Series | bₙ₊₁ ≤ bₙ and lim bₙ = 0 | Conditions not met | Series has form Σ(-1)ⁿbₙ with bₙ > 0 |
The 9 rows of the CalcRef convergence-test reference table.
CalcRef is an educational reference. Check every statement on this sheet against your own course notes, textbook or exam formula policy before relying on it in graded work. A test's hypotheses must hold before its conclusion applies — the divergence test in particular can never establish convergence.