Maclaurin Series -- From Wolfram MathWorld

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A Maclaurin series is a Taylor series expansion of a function about 0,

f(x)=f(0)+f^'(0)x+(f^('')(0))/(2!)x^2+(f^((3))(0))/(3!)x^3+...+(f^((n))(0))/(n!)x^n+.... (1)

Maclaurin series are named after the Scottish mathematician Colin Maclaurin.

The Maclaurin series of a function f(x) up to order n may be found using Series[f, {x, 0, n}]. The nth term of a Maclaurin series of a function f can be computed in the Wolfram Language using SeriesCoefficient[f, {x, 0, n}] and is given by the inverse Z-transform

a_n=Z^(-1)[f(1/x)](n). (2)

Maclaurin series are a type of series expansion in which all terms are nonnegative integer powers of the variable. Other more general types of series include the Laurent series and the Puiseux series.

Maclaurin series for common functions include

1/(1-x)=1+x+x^2+x^3+x^4+x^5+... (3)
for -1<x<1 (4)
cn(x,k)=1-1/2x^2+1/(24)(1+4k^2)x^4+... (5)
cosx=1-1/2x^2+1/(24)x^4-1/(720)x^6+... (6)
for -infty<x<infty (7)
cos^(-1)x=1/2pi-x-1/6x^3-3/(40)x^5-5/(112)x^7-... (8)
for -1<x<1 (9)
coshx=1+1/2x^2+1/(24)x^4+1/(720)x^6+1/(40,320)x^8+... (10)
cot^(-1)x=1/2pi-x+1/3x^3-1/5x^5+1/7x^7-1/9x^9+... (11)
dn(x,k)=1-1/2k^2x^2+1/(24)k^2(4+k^2)x^4+... (12)
erf(x)=1/(sqrt(pi))(2x-2/3x^3+1/5x^5-1/(21)x^7+...) (13)
e^x=1+x+1/2x^2+1/6x^3+1/(24)x^4+... (14)
for -infty<x<infty (15)
_2F_1(alpha,beta;gamma;x)=1+(alphabeta)/(1!gamma)x+(alpha(alpha+1)beta(beta+1))/(2!gamma(gamma+1))x^2+... (16)
ln(1+x)=x-1/2x^2+1/3x^3-1/4x^4+... (17)
for -1<x<=1 (18)
ln((1+x)/(1-x))=2x+2/3x^3+2/5x^5+2/7x^7+... (19)
for -1<x<1 (20)
secx=1+1/2x^2+5/(24)x^4+(61)/(720)x^6+(277)/(8064)x^8+... (21)
sechx=1-1/2x^2+5/(24)x^4-(61)/(720)x^6+(277)/(8064)x^8+... (22)
sinx=x-1/6x^3+1/(120)x^5-1/(5040)x^7+... (23)
for -infty<x<infty (24)
sin^(-1)x=x+1/6x^3+3/(40)x^5+5/(112)x^7+(35)/(1152)x^9+... (25)
sinhx=x+1/6x^3+1/(120)x^5+1/(5040)x^7+1/(362880)x^9+... (26)
sinh^(-1)x=x-1/6x^3+3/(40)x^5-5/(112)x^7+(35)/(1152)x^9-... (27)
sn(x,k)=x-1/6(1+k^2)x^3+1/(120)(1+14k^2+k^4)x^5+... (28)
tanx=x+1/3x^3+2/(15)x^5+(17)/(315)x^7+(62)/(2835)x^9+... (29)
tan^(-1)x=x-1/3x^3+1/5x^5-1/7x^7+... (30)
for -1<x<1 (31)
tanhx=x-1/3x^3+2/(15)x^5-(17)/(315)x^7+(62)/(2835)x^9+... (32)
tanh^(-1)x=x+1/3x^3+1/5x^5+1/7x^7+1/9x^9+.... (33)

The explicit forms for some of these are

1/(1-x)=sum_(n=0)^(infty)x^n (34)
cosx=sum_(n=0)^(infty)((-1)^n)/((2n)!)x^(2n) (35)
cos^(-1)x=pi/2-sum_(n=0)^(infty)(Gamma(n+1/2))/(sqrt(pi)(2n+1)n!)x^(2n+1) (36)
coshx=sum_(n=0)^(infty)1/((2n)!)x^(2n) (37)
cot^(-1)x=pi/2-sum_(n=0)^(infty)((-1)^n)/(2n+1)x^(2n+1) (38)
e^x=sum_(n=0)^(infty)1/(n!)x^n (39)
erf(x)=sum_(n=0)^(infty)(2(-1)^n)/(sqrt(pi)(2n+1)n!)x^(2n+1) (40)
_2F_1(alpha,beta;gamma,x)=sum_(n=0)^(infty)((alpha)_n(beta)_n)/((gamma)_n)(x^n)/(n!) (41)
ln(1+x)=sum_(n=1)^(infty)((-1)^(n+1))/nx^n (42)
ln((1+x)/(1-x))=sum_(n=1)^(infty)2/((2n-1))x^(2n-1) (43)
secx=sum_(n=0)^(infty)((-1)^nE_(2n))/((2n)!)x^(2n) (44)
sechx=sum_(n=0)^(infty)(E_(2n))/((2n)!)x^(2n) (45)
sinx=sum_(n=0)^(infty)((-1)^n)/((2n+1)!)x^(2n+1) (46)
sin^(-1)x=sum_(n=0)^(infty)(Gamma(n+1/2))/(sqrt(pi)(2n+1)n!)x^(2n+1) (47)
sinhx=sum_(n=0)^(infty)1/((2n+1)!)x^(2n+1) (48)
sinh^(-1)x=sum_(n=0)^(infty)(P_(2n)(0))/(2n+1)x^(2n+1) (49)
tanx=sum_(n=0)^(infty)((-1)^n2^(2n+2)(2^(2n+2)-1)B_(2n+2))/((2n+2)!)x^(2n+1) (50)
tan^(-1)x=sum_(n=1)^(infty)((-1)^(n+1))/(2n-1)x^(2n-1) (51)
tanhx=sum_(n=1)^(infty)(2^(2n)(2^(2n)-1)B_(2n))/((2n)!)x^(2n-1) (52)
tanh^(-1)x=sum_(n=1)^(infty)1/(2n-1)x^(2n-1), (53)

where Gamma(x) is a gamma function, B_n is a Bernoulli number, E_n is an Euler number and P_n(x) is a Legendre polynomial.

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