2.3.3—Maclaurin series

The Maclaurin series of f is f(0)+f′(0)x+f″(0)x²/2!+… . It is the Taylor expansion about x=0 and is useful when the series converges and a finite truncation gives the required accuracy.

Find a pattern in derivatives or use known expansions, state the interval or radius of convergence when relevant, and estimate the remainder if an error bound is required.

e^x=1+x+x²/2!+x³/3!+…; for small x, retaining terms through x² gives a controlled approximation whose error is of the order of the next term.

A formal power series is not automatically valid for every x, and truncating after a convenient term does not prove a numerical accuracy.