Polinomio de Taylor

Ejemplo: Obtener el polinomio de Taylor de la función $f (x) = \log (x)$ en x=a (a>0).

\begin{matrix} f (x) = \log x & f (a) = \log (a) \\ f' (x) = x^{- 1} & f' (a) = \frac{1}{a} \\ f'' (x) = (- 1) x^{- 2} & f'' (a) = \frac{(- 1)}{a^{2}} \\ f''' (x) = (- 1) (- 2) x^{- 3} & f''' (a) = \frac{(- 1) (- 2)}{a^{3}} \\ f^{(i v} (x) = (- 1) (- 2) (- 3) x^{- 4} & f^{(i v} (a) = \frac{(- 1) (- 2) (- 3)}{a^{4}} \\ ... & ... \\ f^{(n} (x) = (- 1) (- 2) ... (- (n - 1)) x^{- n} & f^{(n} (a) = \frac{{(- 1)}^{n - 1} (n - 1)!}{a^{n}} \end{matrix}

por lo que

\begin{array}{l} T_{n} [\log x; a] = \log (a) + \frac{1}{a} (x - a) - \frac{1}{2! a^{2}} {(x - a)}^{2} + \frac{1 \cdot 2}{3! a^{3}} {(x - a)}^{3} + . \\ .. + \frac{{(- 1)}^{n - 1} (n - 1)!}{n! a^{n}} {(x - a)}^{n} = \\ = \log (a) + \frac{1}{a} (x - a) - \frac{1}{2 a^{2}} {(x - a)}^{2} + \frac{1}{3 a^{3}} {(x - a)}^{3} + ... + \frac{{(- 1)}^{n - 1}}{n a^{n}} {(x - a)}^{n} \end{array}