Maclaurin series and approximation

A Maclaurin expansion matches a function near zero with $f(0)+f\prime(0)x+f\prime\prime(0)x^2/2!+\cdots$. Standard examples include $e^x$, sine, cosine, $\ln(1+x)$ and the binomial expansion. A polynomial truncation is an approximation; the infinite expansion also has a convergence domain.

Why coefficients contain factorials

Suppose f has a power-series representation near zero that can be differentiated term by term. From $f(x)=a_0+a_1x+a_2x^2+\cdots$, substitution at zero after successive derivatives gives $f(0)=a_0$, $f^{\prime}(0)=a_1$, $f^{\prime\prime}(0)=2!a_2$ and $f^{\prime\prime\prime}(0)=3!a_3$. In general $f^{(n)}(0)=n!a_n$, so $a_n=f^{(n)}(0)/n!$.

$$f(x)=f(0)+xf^{\prime}(0)+\frac{x^2}{2!}f^{\prime\prime}(0)+\cdots+\frac{x^n}{n!}f^{(n)}(0)+\cdots.$$

This is the Maclaurin series listed in MF27. The derivation assumes a valid power-series representation; having derivatives of all orders alone does not guarantee equality to the series. State an applicable interval.

Calculate a function using elementary arithmetic

The source asks how $\sin(0.2)$ could be calculated without a computer or calculator. A suitable polynomial approximation reduces evaluation to addition, subtraction, multiplication and division. Its displayed value is $\sin(0.2)=0.19866933\ldots$ (radians). The source uses this calculation to motivate replacing a difficult function evaluation with elementary arithmetic.

The chapter outline names standard expansions of $(1+x)^n$ for rational n, $e^x$, sine, cosine and $\ln(1+x)$; derivation by repeated differentiation (such as $\sec x$), repeated implicit differentiation (such as $y^3+y^2+y=x^2-2x$), and combining standard series (such as $e^x\cos2x$ and $\ln((1+x)/(1-x))$). It also names convergence intervals, approximation and the small-angle forms $\sin x\approx x$, $\cos x\approx1-x^2/2$, $\tan x\approx x$.

The Student copy is incorporated through printed page 92 and the Teacher through printed page 91, including the standard derivations, Examples 1–11 and Self-Practice 1–4. Further techniques remain pending.