Series Convergence Tests: Geometric, p-Series, Alternating, Ratio, Telescoping, Integral, Limit, Root, and Comparison

Nth Term Test (Series Form)

∑ aₙ

Nth Term Test (Diverges)

lim aₙ ≠ 0 or DNE = diverges

Geometric Series (Series Form)

∑ a rⁿ or ∑ a rⁿ⁻¹

Geometric Series (Converges)

|r| < 1

Geometric Series (Diverges)

|r| ≥ 1

Geometric Series (Comment)

S = a / (1 − r)

p-Series (Series Form)

∑ 1 / nᵖ

p-Series (Converges)

p > 1

p-Series (Diverges)

p ≤ 1

Alternating Series (Series Form)

∑ (−1)ⁿ aₙ or (−1)ⁿ⁺¹ aₙ

Alternating Series (Converges)

aₙ decreasing AND lim aₙ = 0

Alternating Series (Reminder Error)

|Rₙ| ≤ aₙ₊₁

Ratio Test (Series Form)

∑ aₙ

Ratio Test (Converges)

lim |aₙ₊₁ / aₙ| < 1

Ratio Test (Diverges)

lim |aₙ₊₁ / aₙ| > 1

Ratio Test (Comment)

lim |aₙ₊₁ / aₙ| = 1 =inconclusive

Telescoping Series (Series Form)

∑ (bₙ − bₙ₊₁)

Telescoping Series (Converges)

Partial sums = finite limit L

Telescoping Series (Formula)

S = b₁ − L

Integral Test (Series Form)

∑ aₙ and aₙ=f(n)≥0 f has to be continuous, positive, decreasing

Integral Test (Converges)

∫₁^∞ f(x) dx converges f has to be continuous, positive, decreasing

Integral Test (Diverges)

∫₁^∞ f(x) dx diverges f has to be continuous, positive, decreasing

Limit Comparison (Series Form)

∑ aₙ and ∑ bₙ, aₙ,bₙ > 0

Limit Comparison (Converges)

lim (aₙ / bₙ) = c > 0 = both converge if ∑ bₙ converges

Limit Comparison (Diverges)

lim (aₙ / bₙ) = c > 0 = both diverge if ∑ bₙ diverges

Root Test (Series Form)

∑ aₙ

Root Test (Converges)

lim ⁿ√|aₙ| < 1

Root Test (Diverges)

lim ⁿ√|aₙ| > 1

Root Test (Comment)

lim ⁿ√|aₙ| = 1 = inconclusive

Direct Comparison (Series Form)

∑ aₙ and ∑ bₙ, aₙ,bₙ > 0

Direct Comparison (Converges)

0 ≤ aₙ ≤ bₙ and ∑ bₙ converges = ∑ aₙ converges

Direct Comparison (Diverges)

0 ≤ bₙ ≤ aₙ and ∑ bₙ diverges = ∑ aₙ diverges