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

Question

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]

$\begingroup$ Look up FindGeneratingFunction. $\endgroup$
– Szabolcs
Apr 16, 2017 at 9:03 — Szabolcs, Apr 16, 2017 at 9:03
$\begingroup$ Possible duplicates: (38128) (133006) $\endgroup$
– Szabolcs
Apr 16, 2017 at 9:05 — Szabolcs, Apr 16, 2017 at 9:05

ZaMoC · Accepted Answer · 2017-04-16 09:22:46Z

2

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)

answered Apr 16, 2017 at 9:13

ZaMoC

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$\begingroup$ 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] $\endgroup$
– zhk
Apr 16, 2017 at 9:26
$\begingroup$ do you know what the output should be? $\endgroup$
– ZaMoC
Apr 16, 2017 at 9:29
$\begingroup$ Something exponential, maybe like this $a+c(1-\exp{bx})$ not sure $\endgroup$
– zhk
Apr 16, 2017 at 9:30
$\begingroup$ 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 $\endgroup$
– ZaMoC
Apr 16, 2017 at 9:38
$\begingroup$ @MMM I think that it would work better if you fed it with more terms. at least x^4 and x^5... $\endgroup$
– ZaMoC
Apr 16, 2017 at 10:09

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