Sviluppi di Taylor

Data una funzione $f$ derivabile (almeno) $n$ volte in $x_0$, si chiama polinomio di Taylor di ordine $n$ associato a $f$ il polinomio
 
$T_n(x) = f(x_0) + \frac{df}{dx}(x_0) \cdot (x – x_0) + \frac{1}{2!} \frac{d^2 f}{dx^2}(x_0) \cdot (x – x_0)^2 + \ldots + \frac{1}{n!} \frac{d^n f}{dx^n}(x_0) \cdot (x – x_0)^n$
 

Tavola degli sviluppi di Taylor delle funzioni elementari per $x \to 0$

 
$e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \ldots + \frac{x^n}{n!} + o(x^n)$
 
$a^x = 1 + x \ln(a) + \frac{x^2}{2} \ln^2(a) + \frac{x^3}{6} \ln^3(a) + \ldots + \frac{x^n}{n!} \ln^n(a) + o(x^n)$
 
$\sin(x) = x – \frac{x^3}{6} + \frac{x^5}{5!} + \ldots + \frac{(-1)^n}{(2n + 1)!} x^{2n + 1} + o(x^{2x + 2})$
 
$\cos(x) = 1 – \frac{x^2}{2} + \frac{x^4}{4!} + \ldots + \frac{(-1)^n}{(2n)!} x^{2n} + o(x^{2n + 1})$
 
$"tg"(x) = x + \frac{x^3}{3} + \frac{2}{15} x^5 + \frac{17}{315} x^7 + \frac{62}{2835} x^9 + o(x^{10})$
 
$"cotg"(x) = \frac{1}{x} – \frac{x}{3} – \frac{x^3}{45} – \frac{2 x^5}{945} + o(x^6)$
 
$"sec"(x) = 1 + \frac{x^2}{2} + \frac{5 x^4}{24} + \frac{61 x^6}{720} + o(x^7)$
 
$"cosec"(x) = \frac{1}{x} + \frac{x}{6} + \frac{7 x^3}{360} + \frac{31 x^5}{15120} + o(x^6)$
 
$"arcsin"(x) = x + \frac{1}{6} x^3 + \frac{3}{40} x^5 + \ldots + \frac{(2n)!}{4^n \cdot (n!)^2 \cdot (2n + 1)} x^{2n + 1} + o(x^{2n + 2})$
 
$"arccos"(x) = \frac{\pi}{2} – x – \frac{1}{6} x^3 – \frac{3}{40} x^5 – \ldots – \frac{(2n)!}{4^n \cdot (n!)^2 \cdot (2n + 1)} x^{2n + 1} + o(x^{2n + 2})$
 
$"arctg"(x) = x – \frac{x^3}{3} + \frac{x^5}{5} + \ldots + \frac{(-1)^n}{2n + 1} x^{2n + 1} + o(x^{2n + 2})$
 
$"arccotg"(x) = \frac{\pi}{2} – x + \frac{x^3}{3} – \frac{x^5}{5} + \ldots + \frac{-(-1)^n}{2n + 1} x^{2n + 1} + o(x^{2n + 2})$
 
$\sinh(x) = x + \frac{x^3}{6} + \frac{x^5}{5!} + \ldots + \frac{x^{2n + 1}}{(2n + 1)!} + o (x^{2n + 2})$
 
$\cosh(x) = 1 + \frac{x^2}{2} + \frac{x^4}{4!} + \ldots + \frac{x^{2n}}{(2n)!} + o(x^{2n+1})$
 
$"tgh"(x) = x – \frac{x^3}{3} + \frac{2}{15} x^5 – \frac{17}{315} x^7 + \frac{62}{2835} x^9 + o(x^{10})$
 
$"cotgh"(x) = \frac{1}{x} + \frac{x}{3} – \frac{x^3}{45} + \frac{2 x^5}{945} + o(x^6)$
 
$"sech"(x) = 1 – \frac{x^2}{2} + \frac{5 x^4}{24} – \frac{61 x^6}{720} + o(x^7)$
 
$"cosech"(x) = \frac{1}{x} – \frac{x}{6} + \frac{7 x^3}{360} – \frac{31 x^5}{15120} + o(x^6)$
 
$"settsinh"(x) = x – \frac{1}{6} x^3 + \frac{3}{40} x^5 + \ldots + (-1)^n \frac{(2n)!}{4^n \cdot (n!)^2 \cdot (2n + 1)} x^{2n + 1} + o(x^{2n + 2})$
 
$"setttgh"(x) = x + \frac{x^3}{3} + \frac{x^5}{5} + \ldots + \frac{x^{2n + 1}}{2n + 1} + o(x^{2n + 2})$ 
 
$\frac{1}{1 – x} = 1 + x + x^2 + x^3 + \ldots + x^n + o(x^n)$
 
$\ln(1 + x) = x – \frac{x^2}{2} + \frac{x^3}{3} + \ldots + \frac{(-1)^{n+1}}{n} x^n + o(x^n)$
 
$(1 + x)^{\alpha} = 1 + \alpha x + \frac{\alpha (\alpha – 1)}{2} x^2 + \frac{\alpha (\alpha – 1) (\alpha – 2)}{6} x^3 + \ldots + ((\alpha),(n)) x^n + o(x^n)$
 
$\sqrt{1 + x} = 1 + \frac{1}{2} x – \frac{1}{8} x^2 + \frac{1}{16} x^3 + \ldots + (-1)^n\frac{(2n – 1)!!}{(2n + 2)!!} x^{n+1} + o(x^{n+1})$
 
$\frac{1}{\sqrt{1 + x}} = 1 – \frac{1}{2} x + \frac{3}{8} x^2 – \frac{5}{24} x^3 + \ldots + (-1)^{n+1} \frac{(2n + 1)!!}{(2n + 2)!!} x^{n+1} + o(x^{n+1})$
 
 

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