Series

From The Essence Bay
Jump to: navigation, search

Trigonometric functions
Hyperbolic functions
Logarithm
Conics
Vector

Derivative
Integral
Series
Ordinary differential equation

Identities

$$\begin{array}{l} \sum_{k = 0}^n k = \frac{n(n + 1)}{2} \\[6pt] \sum_{k = 0}^n k^2 = \frac{n(n + 1)(2n + 1)}{6} \\[2pt] \sum_{k = 0}^n k^3 = \left(\frac{n(n + 1)}{2}\right)^2 \\[6pt] \sum_{k = 1}^n r^k = \frac{r(1 - r^n)}{1 - r} \\[6pt] \sum_{k = 1}^n k \, r^k = \frac{r(1 - r^n)}{(1 - r)^2} - \frac{n \, r^{n+1}}{1-r} \end{array}$$

Taylor series

Taylor series for $f(x)$ at $x = a$ is equal to:
$$\begin{array}{l} \sum_{k = 0}^\infty \frac{f^{(k)}(a)}{k!} (x - a)^k = f(a) + \frac{f'(a)}{1!} (x - a) + \frac{f''(a)}{2!} (x - a)^2 + \cdots \end{array}$$
If $a = 0$ then this is a special case, called Maclaurin series:
$$\begin{array}{l} \sum_{k = 0}^\infty \frac{f^{(k)}(0)}{k!} (x - 0)^k = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \cdots \end{array}$$
Standard Maclaurin series:
$$\begin{array}{ll} e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots & x \in \mathbb{R} \\[4pt] \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots & -1 < x \leqslant 1 \\[4pt] \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots & x \in \mathbb{R} \\[4pt] \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots & x \in \mathbb{R} \\[4pt] (1 + x)^n = 1 + \frac{q}{1!} x + \frac{q(q - 1)}{2!} x^2 + \frac{q(q - 1)(q - 2)}{3!} x^3 + \cdots & -1 < x < 1, \, n \in \mathbb{Q} \\[4pt] \frac{1}{1 - x} = 1 + x + x^2 + x^3 + \cdots & -1 < x < 1 \\ \end{array}$$