Sviluppi di Taylor centrati in 0

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$$\begin{array}{rl}& -e^x=\sum _{k=0}^n\frac{x^k}{k!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots +\frac{x^n}{n!}+o(x^n)\\ & \\ & -\sin (x)=\sum _{k=0}^n\frac{(-1{)}^k\cdot x^{2k+1}}{(2k+1)!}+o(x^{2n+1})=x-\frac{x^3}{6}+\frac{x^5}{120}+\cdots +\frac{(-1{)}^n\cdot x^{2n+1}}{(2n+1)!}+o(x^{2n+1})\\ & \\ & -\cos (x)=\sum _{k=0}^n\frac{(-1{)}^k\cdot x^{2k}}{(2k)!}+o(x^{2n})=1-\frac{x^2}{2}+\frac{x^4}{24}+\cdots +\frac{(-1{)}^n\cdot x^{2n}}{(2n)!}+o(x^{2n})\\ & \\ & -\frac{1}{1-x}=\sum _{k=0}^n{x}^k+o(x^n)=1+x+x^2+x^3+\cdot +x^n+o(x^n)\\ & \\ & -\log (1+x)=\sum _{k=0}^n\frac{(-1{)}^k\cdot x^{2k+1}}{(2k+1)!}+o(x^{2n+1})=x-\frac{x^3}{6}+\frac{x^5}{120}+\cdots +\frac{(-1{)}^n\cdot x^{2n+1}}{(2n+1)!}+o(x^{2n+1})\end{array}$$

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