Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$.
Recall that $A$ can be defined as the set of expressions of the form $\sum_ua_uu$, where $u$ runs over the set monomials in $x_1,x_2,\dots$, and each $a_u$ is in $K$, the multiplication being the obvious one.
The subset $\mathfrak m$ of $A$ defined by the condition $a_1=0$ is easily seen to be a maximal ideal.
Does the series $\sum_{n\ge1}x_n^n$ converge $\mathfrak m$-adically in $A\ ?$
Three remarks:
(1) The series $\sum_{n\ge1}x_n^n$ is $\mathfrak m$-adically Cauchy.
(2) The condition that $\sum_{n\ge1}x_n^n$ converges $\mathfrak m$-adically is equivalent to the condition that, for all $n$, the element $\sum_{i\ge n}x_i^i$ of $A$ is in $\mathfrak m^n$. Sadly I'm unable to prove (or disprove) this condition even for $n=2$.
(3) Of course the underlying question is to know if $A$ is $\mathfrak m$-adically complete, and I would gladly accept an answer proving that $A$ is not $\mathfrak m$-adically complete, even if the convergence of $\sum_{n\ge1}x_n^n$ is not settled.