Maclaurin series

A Maclaurin series is just a Taylor series that has decided to set up camp at the origin. There is no new idea here — it is the special, and most common, case where the center is the point a = 0. Most of the famous expansions you memorize (for e^x, sin x, cos x) are Maclaurin series.

Formally, the Maclaurin series of a smooth function f is sum_{n=0}^infinity f^{(n)}(0)/n! x^n = f(0) + f'(0) x + f''(0)/2! x^2 + ... . Centering at zero simplifies every term: instead of powers of (x - a) you have plain powers of x, and the coefficients are derivatives evaluated at the convenient point 0.

The name honors Colin Maclaurin, though the construction is entirely Taylor's with a = 0. All the same caveats apply: writing the Maclaurin series down does not guarantee it converges to f. The standard exponential and trigonometric series do converge to their functions for all real x, which is exactly why they are so useful, but this is a fact to be proven, not assumed.