Taylor expansion

What is Taylor expansion?

• Given a real-valued function f ∈ C∞(I) differentiable infinitely many times on an open interval I ⊆ R containing a point a, the power series
• ${\sum }_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$
• is called the Taylor series around the point a of the function f. When the Taylor series converges and coincides with f, we say that f is Taylor expandable.
• $\log (1+x)={\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n}{x}^n$ - f(x) = log(1 + x) - f’(x) = 1/(1 + x) - f”(x) = -1 * (1 + x)^(-2) - f'''(x) = -1 * -2 * (1 + x)^(-3) - f^(n)(x) = (-1)^(n+1) (n - 1)! (1 + x)^(-n - a=0
• $\log (1-x)=-\sum _{n=1}^{\infty }\frac{x^n}{n}$ - f(x) = log(1 - x) - f’(x) = -1 / (1 - x) - f”(x) = -1 * -1 * -1 * (1 + x)^(-2) - f'''(x) = -1 * -1 * -1 * -2 * -1 * (1 + x)^(-3) - f^(n)(x) = (n - 1)! (1 + x)^(-n) - a=0
• If we use this to approximate up to the second order, we get
• $\log (1+x)\approx x-\frac{1}{2}{x}^2$
• $\log (1-x)\approx -x-\frac{1}{2}{x}^2$

from de Moivre-Laplace limit theorem

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