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I was curious if there is any method to be able to sum series like this $$\sum _{n=1}^{\infty } (-1)^n n^n$$ or similar $$\sum _{n=1}^{\infty } n^n (-z)^n$$ for any value os z , I see solutions using integrating Lambert function but it is too complicate I try using levin transform and pade but do not work . it would be convenient to have some general method to sum divergent series like $$\sum _{n=1}^{\infty } (-1)^n n^n \Gamma (n+1)=0.530056382620967931234158702$$ without the use of integrals if possible in this case

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  • $\begingroup$ The sum certainly converges in the ring of $p$-adic integers where we set $\sum_{n=1}^{\infty}(-1)^{n}n^n=\lim_{v>0,v\rightarrow 0}\sum_{n=1}^{\infty}(-1)^{n}n^n$ where the limit and the notion of v converging to $0$ are both taken in the topology of $p$-adic integers. Did you want to restrict your attention to the real or complex numbers, or would a $p$-adic interpretation also be suitable? $\endgroup$ May 10, 2022 at 23:01
  • $\begingroup$ hi@Joseph Van I was interested in acceleration or extrapolation methods that can sum these kinds of divergent sums without the need to transform the sums into integrates where it is possible. $\endgroup$
    – capea
    May 11, 2022 at 7:43

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