Taylor series

Taylor series are common types of power series. Taylor polynomials are partial series of the power series, and are supposed to improve off our understanding of linear approximations. By representing complicated functions with Taylor polynomials we have a substantially easier time in computations. These approximations are centred at a particular point.

$$P_n(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\dots$$

Maclaurin series are a type of Taylor series centred at $a=0$.

$$P_n(x)=f(0)+f^{\prime }(0)x+\frac{f^{\prime \prime }(0)x^2}{2!}+\dots$$

Multivariable case

For a multivariable function, we of course get a far more complicated expansion:

$$z=f(x,y)=f(a,b)+\frac{\partial f}{\partial x}(x-a)+\frac{\partial f}{\partial y}(y-b)+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x^2}(x-a{)}^2+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x\partial y}(x-a)(y-b)+\frac{1}{2}\frac{{\partial }^{2}f}{\partial y\partial x}(y-b)(x-a)+\frac{1}{2}\frac{{\partial }^{2}f}{\partial y^2}(y-b{)}^2+\dots$$

where the partials are evaluated at the point $(a,b)$. That was a long expansion! Remember by Clairaut’s theorem that mixed partials are equivalent, so that provides some simplification relief.

Observe that the first-order Taylor expansion for a multivariable function is the tangent plane.

See also

• Taylor polynomial errors