1 Taylor Series
The Taylor series expansion of a function is an infinite sum of terms that are expressed in terms of the functions derivatives at a single point. A Taylor series is also known as a Maclaurin series when \(0\) is the point where the derivatives are considered. The partial sum formed by the first \(n+1\) terms of a Taylor series is a polynomial of degree \(n\) and is called the \(n\)th Taylor polynomial of the function.
1.1 Taylor Series
The Taylor series of a real or complex function \(f(x)\) that is infinitely differentiable at a real or complex number \(a\) is the power series \[ f(a)+ \frac{f'(a)}{1!}(x-a)+ \frac{f''(a)}{2!}(x-a)^2+ \dots=\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}, \] where \(f^{(n)}(a)\) denotes the \(n\)th derivative of the function \(f\) evaluated at point \(a\).
The Taylor series for a function \(f\) at \(a=0\) is known as the Maclaurin series.
1.2 Taylor Series Examples
1.2.1 1. Exponential Function
The exponential function \(e^x\) has Maclaurin series \[ e^x=\sum_{n=0}^\infty \frac{x^n}{n!}. \] The exponential generating function of the [[bell-numbers|Bell numbers]] is the exponential function of the predecessor of the exponential function \[ \exp(\exp x -1) = \sum_{n=0}^\infty \frac{B_{n}}{n!}x^n. \] ### 2. Natural Logarithm
The natural logarithm has Maclaurin series \[ \begin{align}\log(1-x) & =-\sum_{n=1}^\infty \frac{x^n}{n} \\ \log(1+x) & = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n}.\tag{$\star$}\end{align} \] The series \((\star)\) is known as the Mercator series. Both of these series converge for \(|x|<1\) (in addition, the series for \(\log(1-x)\) converged for \(x=-1\) and the series for \(\log(1+x)\) converges for \(x=1\)).
1.2.2 3. Geometric Series
The geometric series and its derivatives have Maclaurin series \[ \begin{align}\frac{1}{1-x} & = \sum_{n=0}^\infty x^n \\\frac{1}{(1-x)^2} & = \sum_{n=1}^\infty nx^{n-1} \\\frac{1}{(1-x)^3} & = \sum_{{n=2}}^\infty \frac{(n-1)n}{2}x^{n-2} \\& \vdots\end{align} \] All are convergent for \(|x|<1\). These are special cases of the binomial series given below.
1.2.3 4. Binomial Series
The binomial series is the power series \[ (1+x)^\alpha =\sum_{n=0}^\infty {\alpha \choose n}x^n, \] whose coefficients are the generalized binomial coefficients \[ {\alpha\choose n}=\prod_{k=1}^n \frac{\alpha-k+1}{k}=\frac{{\alpha(\alpha-1)\cdots (\alpha-n+1)}}{n!}. \] It converges for \(|x|<1\) for any real or complex number \(\alpha\).
The special cases \(\alpha=\frac{1}{2}\) and \(\alpha= -\frac{1}{2}\) give the square root functon and the inverse square root function \[ \begin{align} (1+x)^{1/2} & = \sum_{n=0}^\infty \frac{(-1)^{n-1}(2n)!}{4^n(n!)^2(2n-1)}x^n \\ (1+x)^{-1/2} & = \sum_{n=0}^\infty \frac{(-1)^n (2n)!}{4^n(n!)^2}x^n. \end{align} \] ### 5. Trigonometric Functions
The usual trigonometric functions and their inverses have the following Maclaurin series \[ \begin{align} \sin x & =\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\ \cos x & = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n} \\ \tan x & = \sum_{n=1}^\infty \frac{B_{2n}(-4)^n(1-4^n)}{(2n)!}x^{2n-1} \\ \sec x & = \sum_{n=0}^\infty\frac{(-1)^nE_{2n}}{(2n)!}x^{2n} \\ \arcsin x& = \sum_{n=0}^\infty \frac{(2n)!}{4^n(n!)^2(2n+1)}x^{2n+1} \\ \arccos x & = \frac{\pi}{2}-\arcsin x \\ \arctan x & = \sum_{n=0}\infty\frac{(-1)^n}{2n+1}x^{2n+1}. \end{align} \] All angles are expressed in radians. The numbers \(B_{k}\) are Bernoulli Numbers and \(E_{k}\) are Euler Numbers.
1.2.4 6. Hyperbolic Functions
The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions \[ \begin{align} \sinh x & = \sum_{n=0}^\infty\frac{x^{2n+1}}{(2n+1)!} \\ \cosh x & = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!} \\ \tanh x & = \sum_{n=1}^\infty\frac{B_{2n}4^n(4^n-1)}{(2n)!}x^{2n-1} \\ \text{arsinh} x & = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{4^n(n!)^2(2n+1)}x^{2n+1} \\ \text{arctanh} & = \sum_{n=0}^\infty \frac{x^{2n+1}}{2n+1}. \end{align} \]