Taylor Series

The Taylor series of a function f (x), that is infinitely differentiable at number a, is the power series:

T (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (a)}{n!} {(x - a)}^{n}

or in other form with (n-1) term and the remainder:

T (x) = \sum_{k = 0}^{n - 1} \frac{f^{(k)} (a)}{k!} {(x - a)}^{k} + R_{n}

Lagrange's form

R_{n} = \frac{f^{(n)} (ξ) {(x - a)}^{n}}{n!}

Cauchy's form

R_{n} = \frac{f^{(n)} (ξ) {(x - ξ)}^{n - 1} (x - a)}{(n - 1)!}

Taylor Serie of some elementary functions

Exponential and Logarithmic functions

e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}; x \in ℝ

\ln (1 + x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n + 1} x^{n + 1}; |x| < 1

Geometric series

\frac{1}{1 - x} = \sum_{n = 0}^{\infty} x^{n}; |x| < 1

\frac{x^{m}}{1 - x} = \sum_{n = m}^{\infty} x^{n}; |x| < 1

Binomial series

{(1 + x)}^{α} = \sum_{n = 0}^{\infty} (\begin{matrix} α \\ n \end{matrix}) x^{n}; |x| < 1 \land α \in ℂ

Trigonometric functions

\sin x = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} x^{2 n + 1}; x \in ℝ

c o s x = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{2 n!} x^{2 n}; x \in ℝ

Hyperbolic functions

s h x = \sum_{n = 0}^{\infty} \frac{1}{(2 n + 1)!} x^{2 n + 1}; x \in ℝ

c h x = \sum_{n = 0}^{\infty} \frac{1}{2 n!} x^{2 n}; x \in ℝ

Keywords: taylor series, remainder, Lagrange

Important functions
Linear polynomial function
Quatratic polynomial function
Cubic polynomial function
Rational function
Root function with even radical
Root function with odd radical
Exponential function
Logarithmic function
Sine function
Cosine function
Tangent function
Cotangent function
Absolute value functions
Function Transformations
Variable and function value transformations
Limits
Limit of a function
Dervation
Table of Derivatives
Differentiation rules
Taylor Series
Integration
Table of Integrals
Integration Rules
Definite Integrals