2.4 Taylor Series

Definition. The Taylor series of ƒ(x), that is infinitely differentiable at a real or complex number a, is the power series

ƒ(x)=ƒ(a)+ƒ(a)′*(x-a)+(ƒ^)(a)′′/2!*(x-a)2+⋯+(ƒ^n)(a)/n!*(x-a)n

Maclaurin Series: Taylor series at a=0:

ƒ(x)=ƒ(0)+(ƒ^′)(0)*x+(ƒ^′*′)(0)/2!*x2+(ƒ^′*″)(0)/3!*x3+⋯=(∑_n=0^∞)((ƒ^n)(0)/n!*xn)

Taylor Series of known functions.

1/(1-x)=1+x+x2+x3+⋯=(∑_n=0^∞)(xn)

ℇx=1+x+(x2)/2!+(x3)/3!+⋯=(∑_n=0^∞)((xn)/n!)

cos(x)=1-(x2)/2!+(x4)/4!-(x6)/6!+⋯=(∑_n=0^∞)((-1)n⋅(x(2*n))/(2*n)!)

sin(x)=x-(x3)/3!+(x5)/5!-(x7)/7!+⋯=(∑_n=0^∞)((-1)(n+1)⋅(x(2*n+1))/(2*n+1)!)

ln(1+x)=x-(x2)/2!+(x3)/3!-(x4)/4!+(x5)/5!-(x6)/6!+(x7)/7!+⋯=(∑_n=0^∞)((-1)(n+1)⋅(xn)/n!)

1/tan(x)=x-(x3)/3!+(x5)/5!-(x7)/7!+⋯=(∑_n=0^∞)((-1)n(x(2*n+1))/(2*n+1)!)

More Cheat Sheets ->