Calculus 2: Series and Sequences Tests

Telescoping Series

Σaₙ = 1 - ½ + ½ - 1/3 + 1/3 - ¼ + … + aₙ

Terms will cancel out. Add remaining terms

Σaₙ = limₙ→∞ Sₙ = L —> Converges

= ±∞/DNE —> Diverges

Geometric Series Test

Σarⁿ⁻¹

|r| < 1 —> Converge

|r| ≥ 1 —> Diverge

(First Term)/1 - r

(Tests for Absolute Convergence)

nᵗʰ Term Test or Divergence Test

limₙ→∞ aₙ ≠ 0 —> Diverges

= 0 —> Inconclusive (Requires More Tests)

Integral Test

aₙ = f(n); f must be positive, continuous, and decreasing [N, ∞]

(From 1 to ∞) —>∫f(x)dx = L —> Converges

= ±∞/DNE —> Diverges

p-Series Test

Σ1/nᵖ (Must be Positive)

p > 1 —> Converges

p ≤ 1 —> Diverges

Direct Comparison Test (DCT)

(Must Be Positive)

0 ≤ aₙ ≤ bₙ

If Σbₙ —> Converges, Σaₙ —> Converges

If Σaₙ —> Diverges, Σbₙ —> Diverges

Limit Comparison Test (LCT)

(Must Be Positive)

limₙ→∞ (aₙ/bₙ) = L

If Σaₙ —> Converges, Σbₙ —> Converges

If Σaₙ —> Diverges, Σbₙ —> Diverges

Alternating Series Test (AST)

Σaₙ(-1)ⁿaₙ —> Converges

1.) limₙ→∞ aₙ = 0

2.) aₙ₊₁ ≤ aₙ

(Must be alternating, decreasing, and the limₙ→∞ aₙ = 0)

Ratio Test

limₙ→∞ |aₙ₊₁/aₙ| < 1 —> Converges

>^1/∞ —> Diverges

= 1 —> Inconclusive (Requires More Tests)

(Tests for Absolute Converg

Root Test

limₙ→∞ ⁿ√|aₙ| < 1 —> Converges

> 1/∞ —> Diverges

= 1 —> Inconclusive (Requires More Tests)

(Tests for Absolute Convergence)

Maclaurin Series of Sin(x)

x - x³/3! + x⁵/5! - x⁷/7! + …

(n=1)∑(-1)ⁿ⁺¹(x)^(2n-1)/(2n-1)!

Maclaurian Series of Cos(x)

1 - x²/2! + x⁴/4! - x⁶/6! + …

(n=0)∑(-1)ⁿ(x)⁽²ⁿ⁾/(2n)!

Maclaurian Series of e^x

1 + x + x² + x³ + x⁴ + … + xⁿ

∑xⁿ/n!

Maclaurian Series of Tan^-1(x)

1 - x³/3 + x⁵/5 - x⁷/7 + …

(n=0)∑(-1)ⁿ(2x)⁽²ⁿ⁺¹⁾/(2n+1), |2x| < 1

Maclaurian Series of 1/(1-x)

1 + x + x² + x³ + … + xⁿ

(n=0)∑xⁿ, |x| < 1

Maclaurian Series of ln(1+x)

x - x²/2 + x³/3 - x⁴/4 + …

(n=1)∑(-1)ⁿ⁺¹(x)ⁿ/n