The following table of Maclaurin expansions summarizes our results so far, and provides expansions for other series that we have not covered.
e x = ∑ k = 0 ∞ x k k ! = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + … {\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 }
sin ( x ) = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! x 2 k + 1 = x − x 3 3 ! + x 5 5 ! − … {\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 }
cos ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k ( 2 k ) ! = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + … {\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 }
sinh ( x ) = ∑ k = 0 ∞ x 2 k + 1 ( 2 k + 1 ) ! = x + x 3 3 ! + x 5 5 ! + x 7 7 ! + … {\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 }
cosh ( x ) = ∑ k = 0 ∞ x 2 k ( 2 k ) ! = 1 + x 2 2 ! + x 4 4 ! + x 6 6 ! + … {\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 }
