Calc Series Test

Test for divergence

divergence: lim(n→∞) aₙ≠0

cannot show convergence

Geometric series test (arⁿ)

convergence: |r|<1

divergence: |r|≥1

Geometric series sum (arⁿ)

a/(1-r)

Telescoping series test (bₙ-bₙ₊₁)

convergence: lim(n→∞)bₙ=L≠∞ (basically do the terms cancel each other out)

divergence: can’t show divergence

Telescoping series sum (bₙ-bₙ₊₁)

b-L

p-series test (1/n^p)

converge: p>1

diverge: p≤1

Alternating series test (-1)ⁿ⁻¹aₙ

converge: 0≤aₙ₊₁≤aₙ AND lim(n→∞)aₙ=0

cannot show divergence

Integral test requirements

ƒ is continuous, positive, and decreasing

Integral test

converges: ∫ (1→∞) f(x)dx converges

diverges: ∫ (1→∞) f(x)dx diverges

Root test

converges: lim(n→ ∞)|aₙ|^^1/n <1

diverges: lim(n→∞)|aₙ|^^1/n >1

inconclusive if it equals 1

Ratio test

converges: lim(n→∞)|(a_n+1)/(a_n)| <1

diverges: lim(n→∞)|(a_n+1)/(a_n)| >1

inconclusive if it equals 1

Direct comparison

converges: 0≤aₙ≤bₙ AND ∑bₙ converges

diverges: 0≤bₙ≤aₙ AND ∑bₙ diverges

Limit comparison test

converges: lim(n→∞) (aₙ/bₙ)=L>0 AND bₙ converges

diverges: lim(n→∞) (aₙ/bₙ)=L>0 AND bₙ diverges

Taylor series for f(x)=1/(1-x)

∑(n=0 to ∞) xⁿ

Expanded taylor series for f(x)=1/(1-x)

1+x+x²+x³+…

Taylor series for f(x)=ln(1-x)

-∑(n=1 to ∞) (xⁿ/n)

Expanded taylor series for f(x)=ln(1-x)

-x-x²/2-x³/3-x⁴/4-…

taylor series for f(x)=arctan(x)

∑(for n=0 to ∞) (-1)ⁿ((x²ⁿ⁺¹)/(2n+1))

expanded taylor series for f(x)=arctan(x)

x-x³/3+x⁵/5-x⁷/7+…

taylor series for f(x)=e^x

∑(for n=0 to ∞) xⁿ/n!

expanded taylor series for f(x)=e^x

1+x+x²/2+x³/3!+x⁴/4!+…

taylor series for f(x)=cos(x)

∑(n=0 to ∞)(-1)ⁿ((x²ⁿ)/(2n)!)

expanded taylor series for f(x)=cos(x)

1-x²/2+x⁴/4!-x⁶/6!+x⁸/8!-…

taylor series for f(x)=sin(x)

∑(n=0 to ∞) (-1)ⁿ((x²ⁿ⁺¹)/(2n+1)!)

expanded taylor series for f(x)=sin(x)

x-x³/3!+x⁵/5!-x⁷/7!+…

coefficients in a Taylor series

(ƒ⁽ⁿ⁾(a))/n!

definition of a taylor series

∑((ƒ⁽ⁿ⁾(a))/n!)(x-a)ⁿ