The Taylor series generated by $f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ is
$\sum_{n=0}^{\infty} a_{n} x^{n}$ A function defined by a power series $\sum_{n=0}^{\infty} a_{n} x^{n}$
with a radius of convergence $R>0$ has a Taylor series that con-
verges to the function at every point of $(-R, R) .$ Show this by
showing that the Taylor series generated by $f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}$ is
the series $\sum_{n=0}^{\infty} a_{n} x^{n}$ itself.
An immediate consequence of this is that series like
$$x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}+\cdots$$
and
$$x^{2} e^{x}=x^{2}+x^{3}+\frac{x^{4}}{2 !}+\frac{x^{5}}{3 !}+\cdots,$$
obtained by multiplying Taylor series by powers of $x,$ as well as
series obtained by integration and differentiation of convergent
power series, are themselves the Taylor series generated by the
functions they represent.