In my textbook the Lagrange's remainder which is associated with the Taylor's formula is defined as:
$R_{n}(x)= \frac{(x-a)^n}{n!} f^{(n)}(a + \vartheta (x-a))$, for some $\vartheta$ $\in$ <0 ,1>
1. I don't understand what this $\vartheta$ represents, why is it here what it means
Then is said that some function can be represented by Taylor series only if its Lagrange's remainder which is associated with its Taylor's formula goes to 0 as n goes to infinity (limit).
2. Why is that, what would be the intuition behind it.
Then to show that f(x)=$e^x$ can be represented by Taylor series it only says:
$R_{n}(x)=\frac{x^n}{n!}e^{\vartheta x}$, for some $\vartheta$ in <0, 1>
3. Shouldn't there be $e^{(a+\vartheta (x-a))}$ instead of $e^{\vartheta x}$ according to the definition?
$\lim_{x \to \infty}|R_{n}(x)|\leq e^{\vartheta x} \lim_{x \to \infty}\frac{|x|^n}{n!}=0$
4. Same as the third question and why is there $\leq$ sign? Isn't the left hand sind exactly equal to the right hand side that is shouldn't there be a $=$ sign?