Taylor and Maclaurin series · Grade 12

In Grade 12 you learn: Maclaurin series.

Many functions can be written as an infinite polynomial, called a power series. The first few terms give an approximation that is excellent near the expansion point. This is how calculators work out sin, eˣ and ln. It is also how physicists simplify equations for small angles.

Key facts

• Maclaurin series: f(x) = f(0) + f′(0)x + (f″(0))/(2!) x^2 + (f‴(0))/(3!) x^3 + …
• e^x = 1 + x + (x^2)/(2!) + (x^3)/(3!) + … for all x.
• sin x = x − (x^3)/(3!) + (x^5)/(5!) − … and cos x = 1 − (x^2)/(2!) + (x^4)/(4!) − … (x in radians).
• ln(1 + x) = x − (x^2)/2 + (x^3)/3 − … for −1 < x ≤ 1. Also (1 + x)^n = 1 + nx + (n(n − 1))/(2!) x^2 + … for |x| < 1.
• Substitute into known series (x → 2x, x → −x², …) rather than differentiating from scratch.