Maclaurin Series and Taylor Polynomials

1/(1-x)

=1+x+x²+... +xⁿ+...

=∑xⁿ

(|x|<1)

1/(1+x)

=1-x+x²+... +(-x)ⁿ+...

=∑(-1)ⁿxⁿ

(|x|<1)

e^(x)

=1+x+x²/2!+...+xⁿ/n!+...

=∑xⁿ/n!

(all real x)

sin(x)

=1-x³/3!+x⁵/5!-...+(-1)ⁿ((x²ⁿ⁺¹)/(2n+1)!)+...

=∑(-1)ⁿ((x²ⁿ⁺¹)/(2n+1)!)

(all real x)

cos(x)

=1-x²/2!+x⁴/4!-...+(-1)ⁿ⁻¹(xⁿ/(2n)!)+...

=∑(-1)ⁿ⁻¹(xⁿ/(2n)!)

(all real x)

ln(1+x)

=x-x²/2+x³/3-...+(-1)ⁿ⁻¹(xⁿ/n)+...

=∑(-1)ⁿ⁻¹(xⁿ/n)

(-1

tan⁻¹(x)

=x-x³/3+x⁵/5-...+(-1)ⁿ((x²ⁿ⁺¹)/(2n+1))+...

=∑(-1)ⁿ((x²ⁿ⁺¹)/(2n+1))

(|x|≤1)

Taylor Series generated by ƒ at x=a

ƒ(a)+ƒ'(a)(x-a)+(ƒ''(a)/2!)(x-a)...(ƒⁿ(a)/n!)(x-a)ⁿ

=∑(ƒⁿ(a)/n!)(x-a)ⁿ