Taylor Series

Author

Eric W. Weisstein
December 12, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/TaylorSeries.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TaylorSeries.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

TaylorTerm[f_,{x_,x0_,n_}]:=Module[{x1},Limit[ Derivative[n][Function[x,f]][x1]/n!,x1->x0]
]

Examples

<<DiscreteMath`RSolve`

Series[f[x],{x,a,2}]//TraditionalForm

(1/(1-x))

SeriesTerm[1/(1-x),{x,a,n}]

Series[1/(1-x),{x,a,5}]

cos(x)

FullSimplify[SeriesTerm[Cos[x],{x,a,n}],n∈Integers&&n>0]

Series[Cos[x],{x,a,5}]

(e^x)

SeriesTerm[Exp[x],{x,a,n}]

Series[Exp[x],{x,a,5}]

ln x

SeriesTerm[Log[x],{x,a,n}]//FullSimplify

Series[Log[x],{x,a,5}]

sin(x)

FullSimplify[SeriesTerm[Sin[x],{x,a,n}],n∈Integers&&n>0]

Series[Sin[x],{x,a,5}]

tan(x)

SeriesTerm[Tan[x],{x,a,n}]

Series[Tan[x],{x,a,3}]//FullSimplify

Multivariate

Series[f[x,y],{x,a,2},{y,b,2}]//TraditionalForm