p-adic convergence:
The p-adic power series $\sum \frac{1}{n!} x^n$ is divergent.
But what about the p-adic power series $\sum \frac{(n!)^n}{1+(n!)^{2n}} x^n$?
Does it converge p-adically?
Answer:
we have $a_n=\frac{(n!)^n}{1+(n!)^{2n}} $.
Now,
$\left|\frac{(n!)^{n}}{1+(n!)^{2n}} \right|_p \sim \left|\frac{(n!)^{n}}{1} \right|_p=\left|(n!)^n \right|_p==p^{-n \ \text{ord}_p(n!)} \to 0$ as $ n \to \infty$.
Thus the p-adic power series $\sum \frac{(n!)^n}{1+(n!)^{2n}} x^n$ converges p-adically.
Is my calculation correct?
Please confirm me.
If my calculation is correct, how can I generalise the power series?
That means , how to extend the coefficient $\frac{(n!)^n}{1+(n!)^{2n}}$ into more general form ?