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I was reading this article (http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf) to see some applications of $p$-adic numbers outside mathematics, and came across these two identities:

$\sum_{n=0}^\infty (-1)^n n!(n+2) = 1$ and

$\sum_{n=0}^\infty (-1)^n n!(n^2-5) = -3$.

Since no $p$ was stated, I figured there must be some trick that shows these are the values in any $\mathbb{Q}_p$, but I couldn't find anything by just looking at some partial sums, unless I missed something. Also, where do these kinds of $p$-adic identities come from? It's easy enough to write down a series that converges in $\mathbb{Q}_p$, but writing down one with a nice limit seems a bit harder.

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Before trying to prove these two you could take a look at $\sum_{n \ge 1} n\cdot n! = - 1$, an example of telescopic series. Indeed, we have $$\sum_{n =1}^m n \cdot n! = \sum_{n =1}^m (n+1)! - n! = (m+1)! - 1.$$

There is something mysterious in series of this type. For example Schikhof in his book "Ultrametric calculus" presents three more and all of them converge to an integral limit: $$ \begin{align} \sum_{n=1}^m \frac{n^2(n+1)!}{4^n} & = -4 + \frac{(2+m)(2+m)!}{4^m}, \\ \sum_{n=1}^m n^2(n+1)! & = 2 + (m-1)(m+2)! \\ \sum_{n=1}^m n^5(n+1)! & = 26 + (m^4 - m^3 - 3m^2 + 12m - 13)(m+2)!. \end{align}$$

The first one converges in $\mathbb Q_p$ exactly when $p \neq 2$, others do not require even that. What makes these odd is that you (probably) can't really replace $4$, $2$ or $5$ by any other value and preserve some nice limit. You can find the right hand sides in almost the same manner as I did with $\sum_n n \cdot n!$.

Back to your question, observe that $$ \begin{align} \sum_{n=0}^m (-1)^n \cdot n! \cdot (n^2 - 5) & = -3 + (m+1)! (m-2)(-1)^m, \\ \sum_{n=0}^m (-1)^n \cdot n! \cdot (n+2) & = 1 + (-1)^m(1+m)! \end{align}$$ But $(m+1)!$ tends to $0$ in the $p$-adic sense regardless of our choice of prime number $p$, which proves both of the identities mentioned in Rozikov's article.

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  • $\begingroup$ Thanks for the reference; I'll be sure to check out Schikhof's book. I can certainly show that those formulas for the partial sums hold (say by induction), but how did you come up with them in the first place? They seem mysterious indeed, but is anything concrete known? Given $c \in \mathbb{Z}$, can I create a series so the partial sum $s_n$ is $c$ plus something times a factorial (so it converges in almost all $\mathbb{Q}_p$ to $c$)? $\endgroup$ – Supersingularity Mar 30 '16 at 21:28
  • $\begingroup$ That's not exactly what you are looking for, but please read the article written by Burger and Struppeck in 1996, "Does it really converge?..." @Supersingularity $\endgroup$ – Santiago Apr 3 '16 at 18:55

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