Maclaurin Series

Author

Eric W. Weisstein
November 2, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<DiscreteMath`RSolve`

Term-by-Term Series

Series[1/(1-x),{x,0,10}]

Series[JacobiCN[x,k^2],{x,0,5}]

Series[Cos[x],{x,0,10}]

Series[ArcCos[x],{x,0,10}]

Series[Cosh[x],{x,0,10}]

Series[ArcCosh[1+x]/x^{1/2}/Sqrt[2],{x,0,10}]

Series[ArcCos[x],{x,0,10}]

Series[Cot[x],{x,0,10}]

Series[ArcCot[x],{x,0,10}]

Series[ArcCot[1/x],{x,0,10}]

Series[Coth[x],{x,0,10}]

Series[ArcCoth[1+x]+1/2Log[x],{x,0,10}]

Series[Csc[x],{x,0,10}]

Series[ArcCsc[x-1],{x,0,10}]

Series[Csch[x],{x,0,10}]

Series[JacobiDN[x,k^2],{x,0,5}]

Series[Erf[x]Sqrt[Pi],{x,0,10}]

Series[Exp[x],{x,0,10}]

Series[Log[1+x],{x,0,10}]

Series[Sec[x],{x,0,10}]

Series[ArcSec[1-x],{x,0,10}]

Series[Sech[x],{x,0,10}]

Series[ArcSech[x],{x,0,20}]

Series[Sin[x],{x,0,10}]

Series[ArcSin[x],{x,0,10}]

Series[Sinh[x],{x,0,10}]

Series[ArcSinh[x],{x,0,10}]

Series[JacobiSN[x,k^2],{x,0,5}]

Series[Tan[x],{x,0,10}]

Series[ArcTan[x+1],{x,0,10}]

Series[Tanh[x],{x,0,10}]

Series[ArcTanh[x],{x,0,10}]

Series[JacobiSN[x,k^2],{x,0,5}]

Normal Distribution Function

Φ[x_]:=Erf[x/Sqrt[2]]/2

Series[Φ[x],{x,0,10}]

Normal[((Series[Erf[y],{y,Infinity,15}] /.y->x/Sqrt[2])/2-1/2)Exp[x^2/2]]

Series[Erf[x],{x,Infinity,10}]

Series[Erf[x]/2,{x,Infinity,10}]

Series[Erf[x/2],{x,Infinity,20}]

Analytic Term Series

(1/(1-x))

Sum[(x^n),{n,0,∞}]

GeneratingFunction[1,n,x]

SeriesCoefficient[1/(1-x),{x,0,n}]

Sum[Piecewise[{{1, n ≥ 0}}, 0](x^n),{n,-∞,∞}]

cosx

V5

SeriesTerm[Cos[x],{x,0,2n}]

Simplify[%,n∈Integers]

V7

FullSimplify[SeriesCoefficient[Cos[x],{x,0,2n}],n∈Integers]

FullSimplify[SeriesCoefficient[Cos[x],{x,0,n}],n∈Integers]

Sum[((((-1))^n)/((2 n)!))(x^(2n)),{n,0,∞}]

Sum[(((((-1))^n) Cos[((n π)/2)])/(n!))(x^n),{n,0,∞}]

GeneratingFunction[((((-1))^n)/((2 n)!)),n,x]

(cos^(-1))(x)

GeneratingFunction[Piecewise[{{Pi/2, n == 0}, {-(Gamma[n/2]/(n*Sqrt[Pi]*Gamma[(1 + n)/2])), 
    Mod[n, 2] == 1 && n >= 0}}, 0],n,x]

Sum[Piecewise[{{Pi/2, n == 0}, {-(Gamma[n/2]/(n*Sqrt[Pi]*Gamma[(1 + n)/2])), 
    Mod[n, 2] == 1 && n >= 0}}, 0](x^n),{n,-∞,∞}]

Sum[Piecewise[{{Pi/2, n == 0}, {-(Gamma[n/2]/(n*Sqrt[Pi]*Gamma[(1 + n)/2])), 
    Mod[n, 2] == 1 && n >= 0}}, 0](x^n),{n,1,∞}]

Sum[Piecewise[{{Pi/2, n == 0}, {-(Gamma[n/2]/(n*Sqrt[Pi]*Gamma[(1 + n)/2])), 
    Mod[n, 2] == 1 && n >= 0}}, 0](x^n),{n,1,∞,2}]

V5

SeriesTerm[ArcCos[x],{x,0,0}]

SeriesTerm[ArcCos[x],{x,0,2n+1}]

Factor[FullSimplify[%,n∈Integers&&n>0]]

FullSimplify[(π/2)+Sum[-((Gamma[(1/2)+n])/((1+2 n) (Sqrt[π]) n!))(x^(2n+1)),{n,0,∞}]]

Factor[FullSimplify[GammaSimplify[%],n∈Integers&&n>0]]

V7

SeriesCoefficient[ArcCos[x],{x,0,n}]

(π/2)+Sum[-((Pochhammer[(1/2),(1/2) (-1+n)])/(n ((1/2) (-1+n))!))(x^n),{n,1,∞,2}]//FullSimplify

FullSimplify[%]

(π/2)+Sum[-((Gamma[(n/2)])/(n (Sqrt[π]) Gamma[((1+n)/2)]))(x^n),{n,1,∞,2}]//FullSimplify

coshx

Sum[(1/((2 n)!))(x^(2n)),{n,0,∞}]

V5

SeriesTerm[Cosh[x],{x,0,2n}]

Simplify[%,n∈Integers]

Simplify[Sum[(1/((2 n)!))(x^(2n)),{n,0,∞}],x>0]

V7

SeriesCoefficient[Cosh[x],{x,0,n}]

Sum[% (x^n),{n,-∞,∞}]

SeriesCoefficient[Cosh[x],{x,0,2n}]

Refine[%,n∈Integers]//Simplify

Sum[% (x^(2n)),{n,-∞,∞}]

(cot^(-1))(x)

(π/2)-Sum[((((-1))^n)/(2n+1))(x^(2n+1)),{n,0,∞}]//FullSimplify

GeneratingFunction[Piecewise[{{Pi/2, n == 0}, {-(I^(-1 + n)/n), Mod[n, 2] == 1 && n >= 0}}, 0],n,x]

V5

SeriesTerm[ArcCot[x],{x,0,n}]

SeriesTerm[ArcCot[x],{x,0,2n+1}]

(π/2)-Sum[((((-1))^n)/(2n+1))(x^(2n+1)),{n,0,∞}]

FullSimplify[%,x>0]

FullSimplify[Subtract@@{(π/2)-ArcTan[x],ArcCot[x]},x>0]

Table[Subtract@@{(π/2)-ArcTan[x],ArcCot[x]},{x,0,10,.2}]//Chop//Union

V7

SeriesCoefficient[ArcCot[x],{x,0,n}]

Sum[% (x^n),{n,-∞,∞}]

FullSimplify[Refine[SeriesCoefficient[ArcCot[x],{x,0,2n+1}],n∈Integers],n∈Integers]

Sum[-((((-1))^n)/(1+2 n))(x^(2n+1)),{n,0,∞}]//FullSimplify

cscx

V5

SeriesTerm[Csc[x],{x,0,2n-1}]

FullSimplify[%,n∈Integers&&n>0]

Sum[(((((-4))^n) BernoulliB[2 n,(1/2)])/((2 n)!))(x^(2n-1)),{n,0,∞}]

V7

SeriesCoefficient[Csc[x],{x,0,n}]

SeriesCoefficient[Csc[x],{x,0,2n-1}]

(e^x)

Sum[(1/(n!))(x^n),{n,0,∞}]

GeneratingFunction[Piecewise[{{1/n!, n >= 0}}, 0],n,x]

V5

SeriesTerm[Exp[x],{x,0,n}]

V7

SeriesCoefficient[Exp[x],{x,0,n}]

Sum[% (x^n),{n,-∞,∞}]

SeriesCoefficient[(E^x),{x,0,n}]

erf(x)

Sum[((2 (((-1))^n))/((1+2 n) (Sqrt[π]) n!))(x^(2n+1)),{n,0,∞}]

GeneratingFunction[Piecewise[{{(2*I^(-1 + n))/(n*Sqrt[Pi]*((1/2)*(-1 + n))!), Mod[n, 2] == 1 && n >= 0}}, 
  0],n,x]

V5

SeriesTerm[Erf[x],{x,0,2n+1}]

Factor[FullSimplify[%,n∈Integers&&n>0]]

Sum[((2 (((-1))^n))/((1+2 n) (Sqrt[π]) n!))(x^(2n+1)),{n,0,∞}]

V7

SeriesCoefficient[Erf[x],{x,0,n}]

Sum[% (x^n),{n,-∞,∞}]

SeriesCoefficient[Erf[x],{x,0,2n+1}]

Factor//@FullSimplify[Refine[%,n∈Integers],n∈Integers]

Sum[((2 (((-1))^n))/((1+2 n) (Sqrt[π]) n!))(x^(2n+1)),{n,0,∞}]

(Subscript[, 2])(Subscript[F, 1])(a,b;c;x)

GeneratingFunction[((Pochhammer[α,n]Pochhammer[β,n])/(Pochhammer[γ,n]n!)),n,x]

V5

SeriesTerm[Hypergeometric2F1[α,β,γ,x],{x,0,n}]

Sum[((Gamma[n+α] Gamma[n+β] Gamma[γ])/(Gamma[1+n] Gamma[α] Gamma[β] Gamma[n+γ]))(x^n),{n,0,∞}]

FullSimplify[%]

V7

SeriesCoefficient[Hypergeometric2F1[α,β,γ,x],{x,0,n}]//FullSimplify

Sum[((Pochhammer[α,n]Pochhammer[β,n])/(Pochhammer[γ,n]))((x^n)/(n!)),{n,0,∞}]

log(x+1)

GeneratingFunction[Piecewise[{{-((-1)^n/n), n >= 1}}, 0],n,x]

Sum[-((((-1))^n)/n)(x^n),{n,∞}]

V5

SeriesTerm[Log[x+1],{x,0,n}]

Simplify[%,n>1]

V7

SeriesCoefficient[Log[x+1],{x,0,n}]

Sum[% (x^n),{n,-∞,∞}]

log(((x+1)/(1-x)))

V5

SeriesTerm[Log[(1+x)/(1-x)],{x,0,2n-1}]

FullSimplify[%,n>0&&n∈Integers]

Sum[(2/(2n-1))(x^(2n-1)),{n,∞}]

FullSimplify[TrigToExp[%],x>0]

V7

SeriesCoefficient[Log[(1+x)/(1-x)],{x,0,2n-1}]

FullSimplify[SeriesCoefficient[Log[(1+x)/(1-x)],{x,0,2n-1}],n∈Integers&&n≥0]

sec(x)

V5

SeriesTerm[Sec[x],{x,0,2n}]

FullSimplify[%,n∈Integers]

Sum[(((((-1))^n) EulerE[2 n])/((2 n)!))(x^(2n)),{n,0,∞}]

Table[(((((-1))^n) EulerE[2 n])/((2 n)!))(x^(2n)),{n,0,5}]

V7

FullSimplify[SeriesCoefficient[Sec[x],{x,0,n}],n∈Integers]

FullSimplify[SeriesCoefficient[Sec[x],{x,0,2n}],n∈Integers]

sech(x)

V5

SeriesTerm[Sech[x],{x,0,2n}]

Sum[((EulerE[2 n])/((2 n)!))(x^(2n)),{n,0,∞}]

V7

sin(x)

Sum[((((-1))^n)/(Gamma[2+2 n]))(x^(2n+1)),{n,0,∞}]

Sum[((I (I^n) (-1+(((-1))^n)))/(2 n!))(x^n)

V5

SeriesTerm[Sin[x],{x,0,2n+1}]

FullSimplify[%,n∈Integers&&n>0]

Simplify[Sum[((((-1))^n)/(Gamma[2+2 n]))(x^(2n+1)),{n,0,∞}],x>0]

V7

SeriesCoefficient[Sin[x],{x,0,n}]

Sum[% (x^n),{n,-∞,∞}]

FullSimplify[%]

(sin^(-1))(x)

V5

SeriesTerm[ArcSin[x],{x,0,2n+1}]

Factor[FullSimplify[%,n∈Integers&&n>0]]

Sum[((Gamma[(1/2)+n])/((1+2 n) (Sqrt[π]) n!))(x^(2n+1)),{n,0,∞}]

V7

sinh(x)

V5

SeriesTerm[Sinh[x],{x,0,2n+1}]

FullSimplify[%,n∈Integers&&n>0]

Simplify[Sum[(1/(Gamma[2+2 n]))(x^(2n+1)),{n,0,∞}],x>0]

V7

SeriesCoefficient[Sinh[x],{x,0,n}]

Sum[% (x^n),{n,0,∞}]

Sum[((x^n)/(n!)),{n,1,∞,2}]

Sum[((x^(2n+1))/((2n+1)!)),{n,0,∞}]

GeneratingFunction[Piecewise[{{1/n!, Mod[n, 2] == 1 && n >= 0}}, 0],n,x]

GeneratingFunction[1/(2n+1)!,n,x]

(sinh^(-1))(x)

V5

SeriesTerm[ArcSinh[x],{x,0,2n+1}]

FullSimplify[%,n>0&&n∈Integers]

Sum[((LegendreP[2 n,0])/(1+2 n))(x^(2n+1)),{n,0,∞}]

Table[(((x^(1+2 n)) LegendreP[2 n,0])/(1+2 n)),{n,0,3}]

FunctionExpand[LegendreP[2n,0]]

Sum[(((Sqrt[π])/(Gamma[(1/2)-n] Gamma[1+n]))/(1+2 n))(x^(2n+1)),{n,0,∞}]

V7

SeriesCoefficient[ArcSinh[x],{x,0,n}]

Sum[(((I^(-1+n)) Pochhammer[(1/2),(1/2) (-1+n)])/(n ((1/2) (-1+n))!))(x^n),{n,1,∞,2}]//FullSimplify

FullSimplify[(((I^(-1+n)) Pochhammer[(1/2),(1/2) (-1+n)])/(n ((1/2) (-1+n))!)),Mod[n,2]==1]

Sum[((Gamma[(n/2)] Sin[((n π)/2)])/(n (Sqrt[π]) Gamma[((1+n)/2)]))(x^n),{n,1,∞,2}]

GeneratingFunction[((Gamma[(n/2)] Sin[((n π)/2)])/(n (Sqrt[π]) Gamma[((1+n)/2)])),n,x]

FullSimplify[((Gamma[(n/2)] Sin[((n π)/2)])/(n (Sqrt[π]) Gamma[((1+n)/2)]))/.n->2n+1,n∈Integers&&n>0]//Factor

Table[(((((-1))^n) Gamma[(1/2)+n])/((1+2 n) (Sqrt[π]) n!)),{n,10}]

Table[((LegendreP[2n,0])/(2n+1)),{n,10}]

FullSimplify[(((((-1))^n) Gamma[(1/2)+n])/( (Sqrt[π]) n!))-LegendreP[2n,0],n∈Integers]

FullSimplify[FunctionExpand[%],n∈Integers]

tan(x)

V5

SeriesTerm[Tan[x],{x,0,2n+1}]

FullSimplify[%,n∈Integers&&n>0]

Simplify[Sum[(((((-1))^n) (4^(1+n)) (-1+(4^(1+n))) BernoulliB[2+2 n])/(Gamma[3+2 n]))(x^(2n+1)),{n,0,∞}],x>0]

V7

SeriesCoefficient[Tan[x],{x,0,n}]

Sum[% (x^n),{n,0,∞}]

Sum[((I (((2 I))^n) (-1+(((-1))^n)) (-1+(2^(1+n))) BernoulliB[1+n])/((1+n)!))(x^n),{n,∞}]//FullSimplify

Refine[((I (((2 I))^n) (-1+(((-1))^n)) (-1+(2^(1+n))) BernoulliB[1+n])/((1+n)!)),Mod[n,2]==0]

Refine[((I (((2 I))^n) (-1+(((-1))^n)) (-1+(2^(1+n))) BernoulliB[1+n])/((1+n)!)),Mod[n,2]==1]

FullSimplify[-((I (I^n) (2^(1+n)) (-1+(2^(1+n))) BernoulliB[1+n])/((1+n)!))/.n->2n+1,n∈Integers&&n>0]

Sum[(((((-1))^n) (4^(1+n)) (-1+(4^(1+n))) BernoulliB[2+2 n])/(Gamma[3+2 n]))(x^(2n+1)),{n,0,∞}]

FullSimplify[%]

GeneratingFunction[(((((-1))^n) (4^(1+n)) (-1+(4^(1+n))) BernoulliB[2+2 n])/(Gamma[3+2 n])),n,x]//FullSimplify

GeneratingFunction[Piecewise[{{(I*(2*I)^n*(-1 + (-1)^n)*(-1 + 2^(1 + n))*BernoulliB[1 + n])/(1 + n)!, 
    n >= 1}}, 0],n,x]

(tan^(-1))(x)

V5

SeriesTerm[ArcTan[x],{x,0,2n-1}]

FullSimplify[%,n∈Integers&&n>0]

Sum[-((((-1))^n)/(-1+2 n))(x^(2n-1)),{n,∞}]

V7

SeriesCoefficient[ArcTan[x],{x,0,n}]

Sum[% (x^n),{n,0,∞}]

Sum[((I^(-1+n))/n)(x^n),{n,1,∞,2}]

FullSimplify[((I^(-1+n))/n),Mod[n,2]==1]

FunctionExpand[((I^(-1+n))/n),Mod[n,2]==1]

GeneratingFunction[Piecewise[{{I^(-1 + n)/n, Mod[n, 2] == 1 && n >= 0}}, 0],n,x]

tanh(x)

V5

SeriesTerm[Tanh[x],{x,0,2n-1}]

Sum[(((2^(2 n)) (-1+(2^(2 n))) BernoulliB[2 n])/((2 n)!)),{n,∞}]

FullSimplify[%,n∈Integers&&n>0]

Simplify[Sum[(((4^n) (-1+(4^n)) BernoulliB[2 n])/((2 n)!))(x^(2n-1)),{n,∞}],x>0]

SeriesTerm[Tanh[x],{x,0,2n+1}]

Simplify[Sum[(((2^(2+2 n)) (-1+(2^(2+2 n))) BernoulliB[2+2 n])/((2+2 n)!))(x^(2n+1)),{n,0,∞}],x>0]

V7

SeriesCoefficient[Tanh[x],{x,0,n}]

SeriesCoefficient[Tanh[x],{x,0,2n-1}]

(tanh^(-1))(x)

V5

SeriesTerm[ArcTanh[x],{x,0,2n-1}]

FullSimplify[%,n∈Integers&&n>0]

Sum[(1/(2n-1))(x^(2n-1)),{n,∞}]

V7

SeriesCoefficient[ArcTanh[x],{x,0,n}]

Sum[% (x^n),{n,0,∞}]

FullSimplify[SeriesCoefficient[ArcTanh[x],{x,0,2n+1}],n∈Integers]

Sum[((x^n)/n),{n,1,∞,2}]

GeneratingFunction[(1/(1+2 n)),n,x]