Frequently Used Maclaurin Series

Below is a list of some of the most frequently used Maclaurin series that will be critical to memorize, many which we have already seen.

• The Geometric Series: $\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + ...$, for $-1 < x < 1$

• The Derivative of the Geometric Series: $\frac{1}{(1 - x)^2} = \sum_{n=0}^{\infty} nx^{n-1} = x + 2x + 3x^2 + ...$, for $-1 < x < 1$.

• The Antiderivative of the Geometric Series: $-\ln (1 - x) = \sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1} = x + \frac{x^2}{2} + \frac{x^3}{3} + ...$, for $-1 ≤ x < 1$.

• Inverse Tangent Function: $\tan ^{-1} x = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - ...$, for $-1 ≤ x ≤ 1$.

• Euler Exponential Function: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + ...$, for $(-\infty, \infty)$.

• Natural Logarithm: $\ln (1 + x) = \sum_{n=0}^{\infty} (-1)^{n+1} \frac{x^n}{n}$.

• Sine Function: $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$, for $(-\infty, \infty)$.

• Cosine Function: $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2} + \frac{x^4}{4} - ...$, for $(-\infty, \infty)$.