If $f$ is analytic on $\mathbb{R}$, is it necessary that $f = \sum_{n = 0}^{\infty} a_{n} x^{n}$ converges for all $x \in \mathbb{R}$?

Question

I have two questions regarding analyticity. They are pretty easy, and I think I have them correct, but I just want to make sure.

First, regarding the question in the title, I think that it is necessary. By the definition of analyticity, we must have the power series in some radius of the function. So, if it is analytic on all of $\mathbb{R}$, it must be within some radius (and thus convergent) for all of $\mathbb{R}$.

Second, if $f$ is analytic, is it necessary for $\text{exp}(f)$ to be analytic? Pretty sure that this again is necessary since $e^{x}$ is analytic, and the composition of analytic functions is analytic.

Answer 1

Counterexample to question 1: $f(x)=\frac1{1+x^2}$. The Maclaurin series of it does not converge everywhere is because $f$ is a meromorphic function in $\mathbb{C}$.
If $f(z)$ is analytic on $\mathbb{C}$, the series you gave is convergent everywhere.
Your explanation of question 2 is correct.