1.1. Taylor series

Taylor polynomials or Taylor series are approximations of a function near a point \(a\). This holds if \(f\) is a function whose \(n+1^{th}\) derivative exists on an interval containing the point \(a\). The approximation is better the more terms you include and the closer you are to the point \(a\).

(1.1)\[f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+...=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\]

1.1.1. McLaurin series

The McLaurin series are special cases of the Tayolor series, where the series are calculated for a region close to origo, setting \(a=0\):

(1.2)\[f(x)=f(a)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+...=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n\]

1.1.2. Examples of Taylor and McLaurin series

Taylor series for \(sin(x)\):

(1.3)\[sin(x)= sin(a)+cos(a)(x-a)-\frac{sin(a)(x-a)^2}{2!}-\frac{cos(a)(x-a)^3}{3!}+...=\sum_{n=0}^{\infty}\frac{sin^{(n)}(a)}{n!}(x-a)^n\]

McLaurin series for \(sin(x)\):

(1.4)\[\begin{split}\begin{aligned} sin(x)& = sin(0)+cos(0)(x-0)-\frac{sin(0)(x-0)^2}{2!}-\frac{cos(0)(x-0)^3}{3!}+...&=\sum_{n=0}^{\infty}\frac{sin^{(n)}(0)}{n!}(x-0)^n\\ & =x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} \end{aligned}\end{split}\]

You can read more about Taylor polynomials in Adams and Essex [AE18] chapter 4.8 and 9.6.