Definition: Maclaurin Series

The Maclaurin series of a function $f$ that is infinitely differentiable at $0$ is its Taylor series centered at $c = 0$ :

\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots.

Several elementary functions have Maclaurin series that converge to the function for all real $x$ :

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}, \qquad \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \qquad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!},

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad (|x| < 1), \qquad \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} x^n \quad (-1 < x \le 1).

10 viewsMathematics · Single Variable Calculus

The Maclaurin series of a function $f$ that is infinitely differentiable at $0$ is its Taylor series centered at $c = 0$ :

\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots.

Several elementary functions have Maclaurin series that converge to the function for all real $x$ :

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}, \qquad \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \qquad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!},

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad (|x| < 1), \qquad \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} x^n \quad (-1 < x \le 1).

10 viewsMathematics · Single Variable Calculus

The Maclaurin series of a function $f$ that is infinitely differentiable at $0$ is its Taylor series centered at $c = 0$ :

\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots.

Several elementary functions have Maclaurin series that converge to the function for all real $x$ :

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}, \qquad \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \qquad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!},

\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad (|x| < 1), \qquad \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} x^n \quad (-1 < x \le 1).

10 viewsMathematics · Single Variable Calculus