A Maclaurin expansion uses derivatives at zero: f(x)= _n=0infinity f(n)(0)n!xn, wherever this series represents the function. For a requested approximation, retain the required number of terms. **Function** | **Expansion** | **Validity** ex | 1+x+ x22!+ x33!+. s | x in R sin x | x- x33!+ x55!-. s | x in R cos x | 1- x22!+ x44!-. s | x in R arctan x | x- x33+ x55-. s | -1 <= x 1 ln(1+x) | x- x22+ x33-. s | -1<x 1 (1+x)p | 1+px+ p(p-1)2!x2+. s | |x|<1 for the general infinite expansion Here p in Q. A non-negative integer p produces a finite binomial polynomial, valid for every real x. Endpoint validity of a general binomial series needs separate consideration. Trigonometric arguments use radians.

A Maclaurin expansion uses derivatives at zero: \(\mathrm{f}(x)=\sum\limits_{n=0}^{\infty}\dfrac{\mathrm{f}^{(n)}(0)}{n!}x^n\), wherever this series represents the function. For a requested approximation, retain the required number of terms.

Here \(p\in\mathbb Q\). A non-negative integer \(p\) produces a finite binomial polynomial, valid for every real \(x\). Endpoint validity of a general binomial series needs separate consideration. Trigonometric arguments use radians.

Maclaurin Series and Power Series

International Baccalaureate Mathematics: Analysis and Approaches Higher Level