Maclaurin and Taylor series

Each series has its own. ex converges for all x; ln(1+x) only for |x| < 1.

Approximation; integration of awkward functions; deriving identities (e.g. Euler's formula).

f(x) = Σ f⁽ⁿ⁾(0) xⁿ/n! (Maclaurin); around a (Taylor).

Maclaurin: power series at x = 0. Taylor: at general point a. Standard expansions for e^x, sin x, cos x, ln(1+x), (1+x)ⁿ are essential.

Worked examples

e^x = 1 + x + x²/2! + x³/3! + ...

sin x = x − x³/3! + x⁵/5! − ...

ln(1+x) = x − x²/2 + x³/3 − ... (|x| < 1).

Frequently asked questions

Range of convergence?

Each series has its own. e^x converges for all x; ln(1+x) only for |x| < 1.

Use?

Approximation; integration of awkward functions; deriving identities (e.g. Euler's formula).