Maclaurin
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Maclaurin Expansion:

f(x)=f(0)+xf′(0)+x22!f′′(0)+…+xnn!f(n)(0)+…\begin{align*} f(x) = &f(0) + xf'(0) + \frac{x^2}{2!}f''(0) \\ &+ \ldots + \frac{x^n}{n!}f^{(n)}(0) + \ldots \end{align*}
(1+x)n=1+nx+n(n−1)2!x2+…+n(n−1)…(n−r+1)r!xr+…\begin{align*} (1&+x)^n = \\ &1 + nx + \frac{n(n-1)}{2!}x^2 + \ldots \\ &+ \frac{n(n-1)\ldots(n-r+1)}{r!}x^r + \ldots \end{align*}
(∣x∣<1)\Big ( |x| < 1 \Big )
ex=1+x+x22!+x33!+…+xrr!+…\mathrm{e}^x = 1 + x + \frac{x^2}{2!}+ \frac{x^3}{3!} + \ldots + \frac{x^{r} }{r!} + \ldots
(all x)(\textrm{all } x)
sin⁡x=x−x33!+x55!−…+(−1)rx2r+1(2r+1)!+…\begin{align*} \sin x = x &- \frac{x^3}{3!}+ \frac{x^5}{5!} - \ldots \\ &+ \frac{(-1)^r x^{2r+1} }{(2r+1)!} + \ldots \end{align*}
(all x)(\textrm{all } x)
cos⁡x=1−x22!+x44!−…+(−1)rx2r(2r)!+…\begin{align*} \cos x = 1 &- \frac{x^2}{2!}+ \frac{x^4}{4!} -\ldots \\ &+ \frac{(-1)^{r} x^{2r} }{(2r)!} + \ldots \end{align*}
(all x)(\textrm{all } x)
ln⁡(1+x)=x−x22+x33−…+(−1)r+1xrr+…\begin{align*} \ln (1+x) = x &- \frac{x^2}{2} + \frac{x^3}{3} - \ldots \\ &+ \frac{(-1)^{r+1} x^{r} }{r} + \ldots \end{align*}
(−1<x≤1)(-1 < x \leq 1)
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This applet aims to make the list of formulas in List MF26 more digital/mobile-friendly. The content is copyright UCLES and MOE 2015, but any errors made herein are mine alone.
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