Q:

Determining a function from its series

If I have the function $e^x$, I know that Mathematica can expand it using
Series[Exp[x],{x,0,3}]

However, let's say that I have the function, $1+x+x^2/2+x^3/6$. Is there a way for me to input this function into Mathematica and have it tell me that it corresponds to the series of $e^x$?
If there is, can Mathematica do the same thing for more complicated series and tell me that it is, for example, a product of two different functions?

A:

You need at least one more term for this specific example
expr = 1 + x + x^2/2 + x^3/6 + x^4/24;

seq = List @@ expr

(*  {1, x, x^2/2, x^3/6, x^4/24}  *)

t[n_] = FindSequenceFunction[seq, n]

(*  x^(-1 + n)/Pochhammer[1, -1 + n]  *)

Sum[t[n], {n, 1, Infinity}]

(*  E^x  *)

