The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The Taylor series of a one dimensional real or complex function $$f(x)$$ that is infinitely differentiable at the point $$x=a$$ is the power series:

$$f(x) = f(a) + \frac{(x-a)}{1!}\left.\frac{df}{dx}\right|_{x=a} + \frac{(x-a)^2}{2!}\left.\frac{d^2f}{dx^2}\right|_{x=a} + \frac{(x-a)^3}{3!}\left.\frac{d^3f}{dx^3}\right|_{x=a} + \dots = \sum_{n=0}^{\infty} \frac{(x-a)^n}{n!}\left.\frac{d^nf}{dx^n}\right|_{x=a} \\ \label{eq:ts1}$$

An alternative form of the one dimensional Taylor series may be obtained by letting $$x=a$$ such that $$x=a+h$$. Substituting into \eqref{eq:ts1}:

$$f(a+h) = f(a) + \frac{h}{1!}\left.\frac{df}{dx}\right|_{x=a} + \frac{h^2}{2!}\left.\frac{d^2f}{dx^2}\right|_{x=a} + \frac{h^3}{3!}\left.\frac{d^3f}{dx^3}\right|_{x=a} + \dots = \sum_{n=0}^{\infty} \frac{h^n}{n!}\left.\frac{d^nf}{dx^n}\right|_{x=a} \\ \label{eq:ts2}$$

The Taylor series may be generalised to functions of more than one variable. The multivariable Taylor series about the point $$(a_1,\dots,a_k)$$ may be expressed as:

$$f(x_1,\dots,x_k) = \sum_{j=0}^{\infty} \left[\frac{1}{j!}\left[\sum_{n=1}^{k} (x_n-a_n)\left.\frac{\partial f}{\partial x_n}\right|_{x_1=a_1,\dots,x_k=a_k} \right]^j\right]\\ \label{eq:ts3}$$

For example, with $$k=2$$:

$$f(x_1,x_2) = f(a_1,a_2) + \left[(x_1-a_1)\left.\frac{\partial f}{\partial x_1}\right|_{(a_1,a_2)} + (x_2-a_2)\left.\frac{\partial f}{\partial x_2}\right|_{(a_1,a_2)} \right] + \frac{1}{2!}\left[(x_1-a_1)\left.\frac{\partial f}{\partial x_1}\right|_{(a_1,a_2)} + (x_2-a_2)\left.\frac{\partial f}{\partial x_2}\right|_{(a_1,a_2)} \right]^2 + \dots \\ \label{eq:ts4}$$ $$\therefore f(x_1,x_2) = f(a_1,a_2) + \left[(x_1-a_1)\left.\frac{\partial f}{\partial x_1}\right|_{(a_1,a_2)} + (x_2-a_2)\left.\frac{\partial f}{\partial x_2}\right|_{(a_1,a_2)} \right] + \frac{1}{2!}\left[(x_1-a_1)^2\left.\frac{\partial^2 f}{\partial {x_1}^2}\right|_{(a_1,a_2)} + 2(x_1-a_1)(x_2-a_2)\left.\frac{\partial^2 f}{\partial x_1x_2}\right|_{(a_1,a_2)} + (x_2-a_2)^2\left.\frac{\partial^2 f}{\partial {x_2}^2}\right|_{(a_1,a_2)}\right] + \dots \\ \label{eq:ts5}$$

As with the one dimensional Taylor series, an alternative form may be obtained by letting $$a_j=x_j-h_j$$ so that $$x_j=a_j+h_j$$. Substituting this into equation \eqref{eq:ts3}:

$$\therefore f(a_1+h_1,a_2+h_2) = f(a_1,a_2) + \left[h_1\left.\frac{\partial f}{\partial x_1}\right|_{(a_1,a_2)} + h_2\left.\frac{\partial f}{\partial x_2}\right|_{(a_1,a_2)} \right] + \frac{1}{2!}\left[(h_1^2\left.\frac{\partial^2 f}{\partial {x_1}^2}\right|_{(a_1,a_2)} + 2h_1h_2\left.\frac{\partial^2 f}{\partial x_1x_2}\right|_{(a_1,a_2)} + h_2^2\left.\frac{\partial^2 f}{\partial {x_2}^2}\right|_{(a_1,a_2)}\right] + \dots \\ \label{eq:ts6}$$

## Example

Express $$y=cos(x)$$ as a Taylor series about the point $$a=0$$.

Determine the function derivatives and evaluate at the point $$a=0$$:

 $$\left.\frac{dy}{dx}\right|_{a=0}=cos(0)=1$$ $$\left.\frac{d^2y}{dx^2}\right|_{a=0}=-sin(0)=0$$ $$\left.\frac{d^3y}{dx^3}\right|_{a=0}=-cos(0)=-1$$ $$\left.\frac{d^4y}{dx^4}\right|_{a=0}=sin(0)=0$$ $$\left.\frac{d^5y}{dx^5}\right|_{a=0}=cos(0)=1$$ $$\left.\frac{d^6y}{dx^6}\right|_{a=0}=-sin(0)=0$$ $$\left.\frac{d^7y}{dx^7}\right|_{a=0}=-cos(0)=-1$$ $$\vdots$$

Substituting the above values into \eqref{eq:ts1}:

$$f(x) = f(0) + \frac{(x-0)}{1!}\left.\frac{df}{dx}\right|_{x=a} + \frac{(x-0)^2}{2!}\left.\frac{d^2f}{dx^2}\right|_{x=a} + \frac{(x-0)^3}{3!}\left.\frac{d^3f}{dx^3}\right|_{x=a} + \dots = 1 - \frac{1}{2!}x^2 + \frac{1}{4!}x^4 - \frac{1}{6!}x^6 + \dots = \sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!} \\ \label{eq:ex1}$$

The figure below shows plots of the Taylor series polynomial for degress of $$n=5,7,9,15,25$$. Observe that as the degree of the polynomial increases, it approaches the correct function. Also note that as the approximation diverges from the point at which it was expanded ($$a=0$$) that the error between the exact function value and the Taylor polynomial series increases.