Maclaurin series,

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