# How to find a function from a series with non-numeric coefficients?

Let us consider a simple example,

Series[Exp[a*x], {x, 0, 5}] // Normal


Now, I want to revert back to Exp[a*x] from it's series expansion.

Is there any such built-in functionality to achieve this?

Thanks to @Szabolcs and @Jenny_mathy for telling me about FindGeneratingFunction, which worked for the above example.

Edit

But for my actual series, I am not getting any output

FindGeneratingFunction[ CoefficientList[a + b x + (c x^2)/2 + 1/6 (b^2 - a c) x^3, x], x]

• Look up FindGeneratingFunction. – Szabolcs Apr 16 '17 at 9:03
• Possible duplicates: (38128) (133006) – Szabolcs Apr 16 '17 at 9:05

I think you want something like this as Szabolcs pointed

FindGeneratingFunction[
CoefficientList[1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120, x], x]


The function changed, so the new one is

FindGeneratingFunction[
CoefficientList[
1 + a x + (a^2 x^2)/2 + (a^3 x^3)/6 + (a^4 x^4)/24 + (a^5 x^5)/120,
x], x]


but it still works

E^(a x)

• But in this case, I am not getting any output. FindGeneratingFunction[ CoefficientList[a + b x + (c x^2)/2 + 1/6 (b^2 - a c) x^3, x], x] – zhk Apr 16 '17 at 9:26
• do you know what the output should be? – J42161217 Apr 16 '17 at 9:29
• Something exponential, maybe like this $a+c(1-\exp{bx})$ not sure – zhk Apr 16 '17 at 9:30
• I tried this FindGeneratingFunction[ CoefficientList[ a + c + b c x + 1/2 b^2 c x^2 + 1/6 b^3 c x^3 + 1/24 b^4 c x^4 + 1/120 b^5 c x^5, x], x] and it works. So the one you are searching must be more complicated – J42161217 Apr 16 '17 at 9:38
• @MMM I think that it would work better if you fed it with more terms. at least x^4 and x^5... – J42161217 Apr 16 '17 at 10:09