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I came across the following sum while studying the integral of $x^x$:

$$\int_0^1 x^x \,\text{d}x = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^n} = 0.783430510712... $$

When I saw this, my first instinct was to think of the Dirichlet Eta Function:

$$\eta({s}) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}$$

However, the term in the exponent of $n$ is, itself, changing with the sum, so this can't be it. I tried looking at other similar zeta functions, but most of them had the same issue.

Is there a defined function that represents the former sum? I couldn't find one through my own research.

Edit:

(1) I now know that the official name for this sum is the Sophomore's Dream. However, I am still wondering if there is a way to represent this function using a zeta function, or something else that is similar. This question's previous title was "Is this sum a variation of a zeta function, or something else?"

(2) Another similar function might be the series expansion of the principal Lambert W Function: $$W_0(x) = \sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}, |x|<\frac1e$$

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  • $\begingroup$ Isn't this related to Somos dream (I hope I remeber the name right) ? $\endgroup$
    – Peter
    Nov 30, 2022 at 16:46
  • $\begingroup$ @Peter I've never heard of that before! Googling that didn't take me to any math-related links. What is that? $\endgroup$
    – Mailbox
    Nov 30, 2022 at 16:46
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    $\begingroup$ This one! Sophomore's dream. Makes sense. $\endgroup$ Nov 30, 2022 at 16:50
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    $\begingroup$ It's the Sophomore's Dream: en.wikipedia.org/wiki/Sophomore%27s_dream $\endgroup$
    – user43208
    Nov 30, 2022 at 16:52
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    $\begingroup$ I know of a more generalized form for the integral in question, and its series expression looks sort of like a Zeta Function. However, it doesn't really answer your question. $\endgroup$ Dec 1, 2022 at 4:14

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