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How can I find the sum of the series

$$ 1+1x + 2! \cdot x^2 + 3!\cdot x^3 + \cdots + n! \cdot x^n $$

I was solving this just out of fun but now it doesn't give away. How to form a general formula for such a series? I have been trying my might and even tried wolfram alpha but it answers me in terms of the complex gamma function and exponential integral function (Ei). Is there a simpler formula and if not how can I derive this huge thing?

MathWorld: Factorial Sums

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  • $\begingroup$ Letting $x=1$, we get the sequence tabulated at oeis.org/A007489 and since there isn't any useful formula given there it's highly unlikely there is one for general $x$. $\endgroup$
    – Gerry Myerson
    Commented Aug 7, 2020 at 6:19
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    $\begingroup$ Mathematica says that Sum[k! x^k, {k, 0, n-1}] is$$(-1/x)e^{-1/x}(E_1(-1/x)-n!x^nE_{n+1}(-1/x))$$ $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 7, 2020 at 6:36
  • $\begingroup$ Should be derivable easily from the recurrence$$nE_{n+1}(z)+zE_n(z)=e^{-z}$$ $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 7, 2020 at 15:58

1 Answer 1

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The problem is not well defined, because it is not clear what exact relation between the series and "an expression" are you looking for. The series is divergent for all $x\neq 0$. However:

It is easy to see that $y(x)=\sum_{n=0}^\infty n!x^n$ satisfies the differential equation $$x^2y'+(x-1)y+1=0,$$ and the initial condition $y(0)=1$. This is a linear first order differential equation, which can be solved explicitly: $$y(x)=x^{-1}e^{-1/x}\left(\int_x^\infty t^{-1}e^{1/t}dt+c\right).$$ Unfortunately, $c$ is arbitrary here, but your series is an asymptotic series at $0$ for all these solutions. Or "formal Taylor series", if you wish. So in some sense your series "represents" all these solutions.

For some classes of moderately diverging series the correspondence between functions and asymptotic series can be made unique. But your series is wildly divergent...

Remark. I recommend the paper by Euler, "De seriebus divergentibus", first published in Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 205-237, reprinted in Opera Omnia: Series 1, Volume 14, pp. 585 - 617, Eneström-Number E247, English translation is available on Euler's web page. There he considers (among many other examples) $$\sum_0^\infty(-1)^nn!x^n$$ but this is slightly less wild than the series you proposed.

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  • $\begingroup$ So that's nothing but the integrating factor times the gamma function plus an arbitrary constant (of integration)?? Thanks anyway, it triggered a new method and approach... Having tried all divergent summation methods out there!!! $\endgroup$
    – user163402
    Commented Aug 9, 2020 at 3:18
  • $\begingroup$ @Prateek Kulkarni: This is not the Gamma function: Gamma does not satisfy ANY algebraic differential equation! Notice that the integrating factor is flat at 0. (All derivatives exist and are equal to zero!). And the integral diverges when $x\to 0$. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 9, 2020 at 3:38
  • $\begingroup$ Oh! I see..Thanks! Actually I am in High School, but reading advanced stuff! out of interest (like finished Higher Algebra, Reading Thomas Calculus' and have finished SL Loney trig both parts!) The one you have attached is an alternating series right...I will check it out, in fact it has a appearance in GH Hardy's "Divergent Series"..Thank you very much for your insight! $\endgroup$
    – user163402
    Commented Aug 9, 2020 at 4:04
  • $\begingroup$ @Prateek Kulkarni: I hope you will enjoy reading Euler. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 9, 2020 at 6:11
  • $\begingroup$ Sir, by any chance is this you?:en.wikipedia.org/wiki/Alexandre_Eremenko $\endgroup$
    – user163402
    Commented Aug 10, 2020 at 7:48

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