AHL 5.19 (HL)—Maclaurin series

A Maclaurin series approximates a function near zero.

The series uses derivatives at 0: f(x)=f(0)+f′(0)x+f″(0)x²/2!+…; truncation creates an approximation error.

Example

e^x≈1+x+x²/2 for small x, so e^.1≈1.105 using three terms.

State the expansion point and order, then judge whether the input lies in a range where the truncation is useful.

A convergent series can still be inaccurate when too few terms are used far from the expansion point.

Standard series: $e^x=1+x+x^2/2!+\cdots$, $\sin x=x-x^3/3!+x^5/5!-\cdots$, $\cos x=1-x^2/2!+x^4/4!-\cdots$, $\ln(1+x)=x-x^2/2+x^3/3-\cdots$, and $(1+x)^p=1+px+p(p-1)x^2/2!+\cdots$. Obtain related series by substituting, multiplying, differentiating or integrating within the valid interval; a differential equation can determine successive coefficients after substituting $y=\sum a_nx^n$.