Maclaurin Series
Maclaurin Series
A Maclaurin expansion uses derivatives at zero: \(\mathrm{f}(x)=\sum\limits_{n=0}^{\infty}\dfrac{\mathrm{f}^{(n)}(0)}{n!}x^n\), wherever this series represents the function. For a requested approximation, retain the required number of terms.
| Function | Expansion | Validity |
|---|---|---|
| \(e^x\) | \(1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots\) | \(x\in\mathbb R\) |
| \(\sin x\) | \(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots\) | \(x\in\mathbb R\) |
| \(\cos x\) | \(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots\) | \(x\in\mathbb R\) |
| \(\arctan x\) | \(x-\dfrac{x^3}{3}+\dfrac{x^5}{5}-\cdots\) | \(-1\le x\le1\) |
| \(\ln(1+x)\) | \(x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\cdots\) | \(-1<x\le1\) |
| \((1+x)^p\) | \(1+px+\dfrac{p(p-1)}{2!}x^2+\cdots\) | \(|x|<1\) for the general infinite expansion |
Here \(p\in\mathbb Q\). A non-negative integer \(p\) produces a finite binomial polynomial, valid for every real \(x\). Endpoint validity of a general binomial series needs separate consideration. Trigonometric arguments use radians.
Why the coefficients contain factorials
For \(P_n(x)=a_0+a_1x+\cdots+a_nx^n\), evaluating its successive derivatives at zero gives \(P_n(0)=a_0\), \(P_n'(0)=a_1\), \(P_n''(0)=2!a_2\), and generally \(P_n^{(k)}(0)=k!a_k\). Matching these values to the derivatives of \(\mathrm f\) determines \(a_k=\mathrm f^{(k)}(0)/k!\).
A finite polynomial is normally an approximation near zero, not an exact formula for the whole function. For example, \(\mathrm e^x\approx1+x+\frac{x^2}{2}\) near zero. Keep an ellipsis when describing the continuing series, and use \(\approx\) when evaluating a truncated approximation.
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