For this question, I will use $sin(x)$ as an example. The Maclaurin series for $\sin(x)$ is $$\sum_{n=0}^{\infty} \frac{{(-1)}^nx^{2n+1}}{(2n+1)!}$$ This gives us a Maclaurin polynomial to represent $\sin(x)$, but is there another way to represent $\sin(x)$ which doesn't give us a polynomial? Is there any case where the Taylor/Maclaurin series gives us a non-polynomial representation for a certain function?
Taylor/Maclaurin Series vs Taylor/Maclaurin Polynomial
calculussequences-and-seriestaylor expansion
Best Answer
There are many ways to represent $\sin\left(x\right)$ which doesn't use a polynomial (of any degree), but with Maclaurin & Taylor series, due to how they're defined, they will always give a "polynomial" (almost always of infinite degree, although polynomials may be defined as only allowing a finite degree, as Eevee Trainer commented to the question) representation for any function.