1.12: Taylor Series
A Taylor series of a function \(f(x)\) about a point \(x = a\) is a power series representation of \(f(x)\) developed so that all the derivatives of \(f(x)\) at \(a\) match all the derivatives of the power series. Without worrying about convergence here, we have \[f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots\nonumber\]
Notice that the first term in the power series matches \(f(a)\), all other terms vanishing, the second term matches \(f'(a)\), all other terms vanishing, etc. Commonly, the Taylor series is developed with \(a = 0\). We will also make use of the Taylor series in a slightly different form, with \(x = x_* + \epsilon\) and \(a = x_*\):
\[f(x_*+\epsilon)=f(x_*)+f'(x_*)\epsilon+\frac{f''(x_*)}{2!}\epsilon^2+\frac{f'''(x_*)}{3!}\epsilon^3+\cdots\nonumber\]
Another way to view this series is that of \(g(\epsilon ) = f(x_* + \epsilon)\), expanded about \(\epsilon = 0\).
Taylor series that are commonly used include \[\begin{aligned} e^x&=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots , \\ \sin x&=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots , \\ \cos x&=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots , \\ \frac{1}{1+x}&=1-x+x^2-\cdots ,\quad\text{for }|x|<1, \\ \ln(1+x)&=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots ,\quad\text{for }|x|<1.\end{aligned}\]