Standard Maclaurin expansions

Some Maclaurin expansions must be known by heart: they are the “building blocks” from which, by substitution and combination, all the others are obtained.

In brief — Fundamental Maclaurin expansions

As $x\to 0$ (Peano remainder):

e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + o(x^n) \\ \sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + (-1)^k\frac{x^{2k+1}}{(2k+1)!} + o(x^{2k+2}) \\ \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + (-1)^k\frac{x^{2k}}{(2k)!} + o(x^{2k+1}) \\ \ln(1+x) &= x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^{n+1}\frac{x^n}{n} + o(x^n) \\ \frac{1}{1-x} &= 1 + x + x^2 + x^3 + \cdots + x^n + o(x^n) \quad (|x|<1) \\ (1+x)^\alpha &= 1 + \alpha x + \frac{\alpha(\alpha-1)}{2}x^2 + \cdots + \binom{\alpha}{n}x^n + o(x^n) \end{aligned}$$

Example — Numerical approximation of $e$

From $e^x = 1 + x + x^2/2 + x^3/6 + \cdots$ with $x = 1$: $e \approx 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} = 2{,}71\overline{6}.$ The true value is $e\approx 2{,}71828$: the error is $\approx 1{,}6\cdot 10^{-3}$. Using the Lagrange remainder with $c\in(0,1)$: $|R_5(1)| = \frac{e^c}{6!}\cdot 1^6 < \frac{e}{720} < \frac{3}{720} \approx 4{,}2\cdot 10^{-3},$ a bound consistent with the observed error.

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Topics: Derivatives
Concepts: Maclaurin polynomial · Taylor remainder · Standard expansion
Methods: Taylor Lagrange remainder
Skills: Estimating · Using formulae
People: Maclaurin