For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?

Question

this is a question from a book I'm struggling with, please could you provide a clear proof

For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?

kind thanks

Answer 1

A series $\sum_{n=0}^\infty a_n$ with $a_n\in\Bbb Q$ (say) converges $p$-adically if and only if $$ |a_n|_p\rightarrow0\qquad \text{as $n\to\infty$} $$ Now $$ v_p(n!)=\left\lfloor\frac np\right\rfloor+\left\lfloor\frac n{p^2}\right\rfloor+ \left\lfloor\frac n{p^3}\right\rfloor+\cdots $$ is increasing and unbounded so that $$ \lim_{n\to\infty}|n!|_p=\lim_{n\to\infty}p^{-v_p(n!)}=0. $$ Thus, the given series converges $p$-adically for all (finite) $p$.