Science:Infinite Series Module/Units/Unit 3/3.2 Taylor Series/3.2.06 Maclaurin Expansions

The following table of Maclaurin expansions summarizes our results so far, and provides expansions for other series that we have not covered.

The Maclaurin Expansions of Elementary Functions

$e^x=\sum _{k=0}^{\mathrm{\infty }}\frac{x^k}{k!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots$ $\sin (x)=\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{(2k+1)!}{x}^{2k+1}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots$ $\cos (x)=\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k{x}^{2k}}{(2k)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots$ $\sinh (x)=\sum _{k=0}^{\mathrm{\infty }}\frac{x^{2k+1}}{(2k+1)!}=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\dots$ $\cosh (x)=\sum _{k=0}^{\mathrm{\infty }}\frac{x^{2k}}{(2k)!}=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\dots$