# 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

${\displaystyle e^{x}=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\ldots }$

${\displaystyle \sin(x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}x^{2k+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\ldots }$

${\displaystyle \cos(x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)!}}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\ldots }$

${\displaystyle \sinh(x)=\sum _{k=0}^{\infty }{\frac {x^{2k+1}}{(2k+1)!}}=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\ldots }$

${\displaystyle \cosh(x)=\sum _{k=0}^{\infty }{\frac {x^{2k}}{(2k)!}}=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\ldots }$