Series Expansion

In mathematics, a series expansion of a function $ f $ in the neighborhood of a point $ a $ is expressed in the form $$ f(x) = P_n(x-a) + O(x^{n+1}) $$ where $ P_n $ is a polynomial of degree less than or equal to $ n $, and $ O(x^{n+1}) $ is a remainder negligible compared to $ (x-a)^n $ in the neighborhood of $ a $

A series expansion of order $ n $ therefore provides the best local polynomial approximation of the function up to the term of degree $ n $. The higher the order, the more accurate the approximation is near $ a $.

To compute a (limited) series expansion of order $ n $ of a function $ f(x) $ in the neighborhood of a value $ a $, if the function is differentiable in $ a $, then it is possible to use the Taylor-Young formula which decomposes any function into:

$$ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f^{(2)}(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^{n} + O(x^{n+1}) \\ = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^{k} + O(x^{n+1}) $$

with $ O(x^n) $ the Big O (Landau's asymptotic notation) indicating precision, a value tending to be negligible with respect to $ (x-a)^n $ in the neighborhood of $ a $.

The following expansions are valid in the neighborhood of $ 0 $, here is a list of common Taylor series expansions to know.

— Exponential function (exp) and logarithm functions (ln or log):

$$ \begin{aligned} \exp(x) &= \sum_{n=0}^{\infty} \frac{x^n}{n!} \\ &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + O(x^n+1) \\ \ln(1-x) &= -\sum_{n=1}^{\infty} \frac{x^n}{n} \\ &= -x- \frac{x^2}{2} - \frac{x^3}{3} - \cdots - \frac{x^n}{n} + O(x^n+1) \\ \ln(1+x) &= \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \\ &= x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^{n+1} \frac{x^n}{n} + O(x^n+1) \end{aligned} $$

— Power and root functions (sqrt):

$$ \begin{aligned} (1+x)^a &= \sum_{n=0}^{\infty}\binom{a}{n} x^n \\ &= \sum_{n=0}^{\infty} x^n \prod _{k=1}^{n}{\frac {\alpha -k+1}{k}} \\ &= 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3 + \cdots + \frac{a(a-1)(a-2)\cdots(a-n+1)}{n!}x^n + O(x^n+1) \\ (1+x)^{1/2} &= \sqrt{1+x} \\ &= 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \frac{7}{256}x^5 + \cdots \\ (1+x)^{-1/2} &= \frac{1}{\sqrt{1+x}} \\ &= 1 -\frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \frac{35}{128}x^4 - \frac{63}{256}x^5 + \cdots \end{aligned} $$

— Inverse functions

$$ \begin{aligned} \frac{1}{1+x} &= \sum_{n=0}^{\infty} (-1)^n x^n \\ &= 1 - x + x^2 - x^3 + \cdots + (-1)^n x^n + O(x^n) \\ \frac{1}{(1+x)^2} &= \sum_{n=0}^{\infty} (-1)^n nx^{n-1} \\ &= 1 - 2x + 3x^2 - \cdots + (-1)^n nx^{n-1} + O(x^n) \\ \frac{1}{1-x} &= \sum_{n=0}^{\infty} x^{n} \\ &= 1 + x + x^2 + \cdots + x^n + O(x^n) \\ \frac{1}{(1-x)^2} &= \sum_{n=1}^{\infty} nx^{n-1} \\ &= 1 + 2x + 3x^2 + \cdots + nx^{n-1} + O(x^n) \end{aligned} $$

— Trigonometric functions (cosine, sine, tangent)

$$ \begin{aligned} \cos(x) &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} \\ &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + \frac{(-1)^n}{(2n)!} x^{2n} + O(x^{2n+1}) \\ \sin(x) &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} \\ &= x - \frac{x^3}{3!} + \frac{x^{5}}{5!} - \cdots + \frac{(-1)^n}{(2n+1)!} x^{2n+1} + O(x^{2n+2}) \\ \tan(x) &= \sum^{\infty}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} \\ &= x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots + \frac{B_{2n}(-4)^n(1-4^n)}{(2n)!} x^{2n-1} + O(x^{2n}) \\ \sec(x) &= \sum^{\infty}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n} \\ &= 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \cdots + \frac{(-1)^n E_{2n}}{(2n)!} x^{2n} + O(x^{2n+1}) \end{aligned} $$

with $ E_n $ the Euler numbers.

— Complementary and reciprocal trigonometric functions

$$ \begin{aligned} \arccos(x) &= \frac{\pi}{2} - \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} \\ &= \frac{\pi}{2} - x - \frac{x^3}{2 \times 3} - \frac{1 \times 3 \times x^5}{2 \times 4 \times 5} - \cdots - \frac{1 \times 3 \times 5 \cdots (2n-1)x^{2n+1}}{2 \times 4 \times 6 \cdots (2n) \times (2n+1)} + O(x^{2n+2}) \\ \arcsin(x) &= \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} \\ &= x + \frac{x^3}{2 \times 3} + \frac{1 \times 3 \times x^5}{2 \times 4 \times 5} + \cdots + \frac{1 \times 3 \times 5 \cdots (2n-1)x^{2n+1}}{2 \times 4 \times 6 \cdots (2n) \times (2n+1)} + O(x^{2n+2}) \\ \arctan(x) &= \sum^{\infty}_{n=0} (-1)^{n}\frac{x^{2n+1}}{2n+1} \\ &= x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots + (-1)^{n}\frac{x^{2n+1}}{2n+1} + O(x^{2n+2}) \end{aligned} $$

— Hyperbolic and reciprocal trigonometric functions

$$ \begin{aligned} \cosh(x) &= \sum^{\infty}_{n=0} \frac{x^{2n}}{(2n)!} \\ &= 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots + \frac{x^{2n}}{(2n)!} + O(x^{2n+1}) \\ \sinh(x) &= \sum^{\infty}_{n=0} \frac{x^{2n+1}}{(2n+1)!} \\ &= x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots + \frac{x^{2n+1}}{(2n+1)!} + O(x^{2n+2}) \\ \tanh(x) &= \sum^{\infty}_{n=1} \frac{B_{2n} 4^n \left(4^n-1\right)}{(2n)!} x^{2n-1} \\ &= x - \frac{x^3}{3} + \frac{2x^5}{15} - \frac{17x^7}{315} + \cdots + \frac{B_{2n} 4^n (4^{n}-1)}{(2n)!} x^{2n-1} + O(x^{2n}) \\ \operatorname{asinh}(x) &= \sum^{\infty}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} \\ &= x - \frac{x^3}{2 \times 3} + \cdots +(-1)^{n} \frac{1 \times 3 \times 5 \cdots (2n-1)x^{2n+1}}{2 \times 4 \times 6 \cdots (2n) \times (2n+1)} + O(x^{2n+2}) \\ \operatorname{atanh}(x) &= \sum^{\infty}_{n=0} \frac{x^{2n+1}}{2n+1} \\ &= x + \frac{x^3}{3} + \cdots + \frac{x^{2n+1}}{2n+1} + O(x^{2n+2}) \end{aligned} $$

with $ B_n $ the Bernoulli numbers

A Taylor series is an infinite series (without approximation).

A Taylor series expansion of order n is a finite polynomial approximation.

In most practical cases, calculating a Taylor series expansion involves using Taylor's formula and stopping at the desired degree.

When the Taylor series converges to the function (as with analytic functions), the Taylor series expansions are equal to the partial sums of the Taylor series.