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+...
for -1<x<1
cn(x,k)=1-1/2x^2+1/(24)(1+4k^2)x^4+...
cosx=1-1/2x^2+1/(24)x^4-1/(720)x^6+...
for -infty<x<infty
cos^(-1)x=1/2pi-x-1/6x^3-3/(40)x^5-5/(112)x^7-...
for -1<x<1
coshx=1+1/2x^2+1/(24)x^4+1/(720)x^6+1/(40,320)x^8+...
cot^(-1)x=1/2pi-x+1/3x^3-1/5x^5+1/7x^7-1/9x^9+...
dn(x,k)=1-1/2k^2x^2+1/(24)k^2(4+k^2)x^4+...
erf(x)=1/(sqrt(pi))(2x-2/3x^3+1/5x^5-1/(21)x^7+...)
e^x=1+x+1/2x^2+1/6x^3+1/(24)x^4+...
for -infty<x<infty
_2F_1(alpha,beta;gamma;x)=1+(alphabeta)/(1!gamma)x+(alpha(alpha+1)beta(beta+1))/(2!gamma(gamma+1))x^2+...
ln(1+x)=x-1/2x^2+1/3x^3-1/4x^4+...
for -1<x<=1
ln((1+x)/(1-x))=2x+2/3x^3+2/5x^5+2/7x^7+...
for -1<x<1
secx=1+1/2x^2+5/(24)x^4+(61)/(720)x^6+(277)/(8064)x^8+...
sechx=1-1/2x^2+5/(24)x^4-(61)/(720)x^6+(277)/(8064)x^8+...
sinx=x-1/6x^3+1/(120)x^5-1/(5040)x^7+...
for -infty<x<infty
sin^(-1)x=x+1/6x^3+3/(40)x^5+5/(112)x^7+(35)/(1152)x^9+...
sinhx=x+1/6x^3+1/(120)x^5+1/(5040)x^7+1/(362880)x^9+...
sinh^(-1)x=x-1/6x^3+3/(40)x^5-5/(112)x^7+(35)/(1152)x^9-...
sn(x,k)=x-1/6(1+k^2)x^3+1/(120)(1+14k^2+k^4)x^5+...
tanx=x+1/3x^3+2/(15)x^5+(17)/(315)x^7+(62)/(2835)x^9+...
tan^(-1)x=x-1/3x^3+1/5x^5-1/7x^7+...
for -1<x<1
tanhx=x-1/3x^3+2/(15)x^5-(17)/(315)x^7+(62)/(2835)x^9+...
tanh^(-1)x=x+1/3x^3+1/5x^5+1/7x^7+1/9x^9+....
The explicit forms for some of these are
1/(1-x)=sum_(n=0)^(infty)x^n SOURCE: MATHWORLD
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