MHB Is the power series convergent in the field of p-adic numbers?

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Here is this week's POTW:

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Find the domain of convergence of the power series $$\sum\limits_{n = 1}^\infty \frac{(-1)^{n-1}x^n}{n}$$ in the field $\Bbb Q_p$ of $p$-adic numbers.

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No one answered this week's problem. Here is my solution below.

Spoiler

Since $(|(-1)^{n+1}/n|_p)^{1/n} = (|1/n|_p)^{1/n} = p^{\frac{\text{ord}_p n}{n}} \to 1$ as $n\to \infty$, by the ratio test, the series converges for $|x|_p < 1$ and diverges for $|x|_p > 1$. When $|x|_p = 1$, $|(-1)^{n+1}x^n/n|_p = p^{\text{ord}_p n} \ge 1$ for all $n$, which implies that the series diverges. Hence, the domain of convergence is the open unit disk in $\Bbb Q_p$ centered at $0$, or alternatively the closed disk in $\Bbb Q_p$ centered at $0$ of radius $1/p$.