Remember that if the Taylor series of a function f ( x ) {\displaystyle f(x)} is given by f ( x ) = ∑ n = 0 ∞ a n x n {\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}} , the Taylor series of f ( x m ) {\displaystyle f(x^{m})} and x m f ( x ) {\displaystyle x^{m}f(x)} are given by f ( x m ) = ∑ n = 0 ∞ a n x n m and x m f ( x ) = ∑ n = 0 ∞ a n x n + m . {\displaystyle {\begin{aligned}f(x^{m})=\sum _{n=0}^{\infty }a_{n}x^{nm}\ {\text{and}}\ x^{m}f(x)=\sum _{n=0}^{\infty }a_{n}x^{n+m}.\end{aligned}}}
