Existence of closed form function for every power series

Question

We know that corresponding to every analytic real valued function of a real variable there is a power series representation. I was just curious if the converse is true or not. The $a_n$ involved in the summation $$\sum_{n=0}^\infty a_nx^n$$ can be any random function of $n$ which could supress all possibilities of a closed form representation. If the answer happens to be true, please provide for sufficient details. Thanks in advance.

Answer 1

If the power series has a positive radius of convergence $R$ then in the interval $(-R,R)$ it defines a real analytic function. That function is unlikely to have a closed form representation if by "closed form" you mean some rational expression involving polynomials and exponential/trigonometric functions.

In fact the term by term integral of your power series will define an analytic function that is even more unlikely to have a closed form expression: see

How can you prove that a function has no closed form integral?

and

https://en.wikipedia.org/wiki/Closed-form_expression