It is well known that the harmonic series $$\sum_{n=1}^\infty \frac{1}{n}$$ is not convergent with standard metric on $\mathbb{R}$, i.e. with $d(x,y)=|x-y|$.
Also the series does not converge in field of $p$-adic numbers $\mathbb{Q}_p$ with $p$-adic absolute value $|\;|_p$ since $$\lim_{n\to\infty}^p \frac{1}{n}\ne 0,$$ where by $\displaystyle{\lim^p}$ I mean taking limit according to $p$-adic absolute value $|\;|_p$.
Well, for any field $K$ of characteristic $0$, $\mathbb{Q}$ (the field of rational numbers) is a subfield of $K$. Thus my question is:
Is there any normed field $K$ such that the harmonic series converges?
No. Since all the partial sums $\displaystyle\sum_{n=1}^N\frac 1n$ are in $\mathbb Q$, the harmonic series is Cauchy in a normed field $(K, \|\cdot\|)$ if and only if it is Cauchy in $(\mathbb Q, \|\cdot\|_\mathbb Q)$, where $\|\cdot\|_\mathbb Q$ is the restriction of the norm on $K$ to $\mathbb Q$. But by Ostrowski's theorem, the only norms on $\mathbb Q$ are the Euclidean one, the trivial one and the $p$-adic ones, in which the harmonic series diverges.