Question:How would you prove the Taylor series for a function, and how would you find the exact value of it?

I know that$$e^x=\sum_{k=0}^{\infty}\frac {x^k}{k!}\\\frac 1{1-x}=\sum_{k=0}^\infty x^k\\\sin x=\sum_{k=0}^\infty \frac {(-1)^k}{(2k+1)!}x^{2k+1}\\\vdots$$

My question is: How would you prove the taylor expansion for any polynomial $f(x)$, and how would you evaluate the convergence?

** For Example:** With $e^x$, letting $x$ be a simple integer like $2$, we have\begin{align*}e^2=1+2+\frac 4{2!}+\frac 8{3!}+\ldots\tag1\end{align*}
How would you solve for the exact value of $e^2$ when you have the infinite sequence? (Note that it

*does*converge) And $x$ doesn't

*have*to be an integer. It can also be a number such as $\pi\sqrt{19}$.\begin{align*}e^{\pi\sqrt{19}}=1+\pi\sqrt{19}+\frac {19\pi^2}{2!}+\frac {19\pi^3\sqrt{19}}{3!}+\frac {19^2\pi^4}{4!}+\ldots\tag2\end{align*} Which is intimating.