Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring? I asked this question on Mathematics Stackexchange (link), but got no answer.
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 addition and multiplication being the obvious ones. 
Then $A$ is a local domain, its maximal ideal $\mathfrak m$ is defined by the condition $a_1=0$, and it seems natural to ask

Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?

I suspect that the answer is No, and that the series $\sum_{n\ge1}x_n^n$, which is clearly Cauchy, does not converge $\mathfrak m$-adically.
 A: [Edit: The lemma was revised  and proved, changed the point of view from series to sequences.]
[2nd Edit: The proof of the lemma was improved, and   now the argument can show that Cauchy series such as   $x_1+(x_{1^3+1}^2+\dots x_{2^3}^2)+(x_{2^3+1}^{3}+\dots+x_{3^3}^3)+\dots$, diverge in the $\mathfrak{m}$-adic topology.]
Here is a construction of a Cauchy sequence which does not converge. It is based on the following lemma, a proof of which will is given at the end.

Lemma. Let $\mathfrak{m}_c$ denote the maximal ideal of $K[[x_1,\dots,x_c]]$.
  Then there exist  sequences of natural numbers $(r_n)_{n\in\mathbb{N}}$, $(c_n)_{n\in\mathbb{N}}$  and a sequence of elements $(p_n\in(\mathfrak{m}_{c_n})^n)_{n\in\mathbb{N}}$ such that:
  
  
*
  
*$\limsup r_n=\infty$ and 
  
*$p_n$ cannot be written as a sum of $r_n$ terms $\sum_{i=1}^{r_n} a_{i} b_i$ with $a_{i},b_i\in\mathfrak{m}_{c_n}$.
  
  
  In fact, one can take $c_n=n^2$, $p_n=x_1^{n}+\dots+x_{n^2}^{n}$ and let $r_n=\lceil \frac{n^2}{2(n-1)}\rceil-1$ if $\mathrm{char}\,K\nmid n$ and  $r_n=1$ when $\mathrm{char}\,K\mid n$.  

It would be more convenient to replace $K[[x_1,x_2,\dots]]$ with the isomorphic ring $$S:=K[[y_{ij}\,|\,i,j\in \mathbb{N}]].$$
For every $n$, there is a ring homomorphism $\phi_n:S\to K[[x_1,\dots,x_{c_n}]]$ specializing $y_{n1},\dots,y_{n{c_n}}$ to $x_1,\dots,x_{c_n}$ and the rest of the variables to $0$.
Let $f_n=p_n(y_{n1},\dots,y_{nc_n})$ and  define in $S$ the (formal) partial sums
$$
g_t:=\sum_{n=t}^\infty f_n.
$$ 
Then, by construction, $(g_t)_{t\in\mathbb{N}}$ is a Cauchy sequence relative to the $\mathfrak{m}$-adic topology, but it does not converge.
Indeed, if $(g_t)_t$  converges in the $\mathfrak{m}$-adic topology, then it also converges relative to the (coarser) grading topology of $S$, and so must converge to $0$. However, if this is so, there exists  $t\in\mathbb{N}$ such that $g_t=\sum_{n=t}^\infty f_n\in \mathfrak{m}^2$. In particular, we can write  $\sum_{n=t}^\infty f_n=\sum_{i=1}^u a_{i}b_i$ with $a_{i},b_i\in \mathfrak{m}$.
Choose $n\geq t$ sufficiently large to have $r_n\geq u$. Applying $\phi_n$ to both sides of the last equality gives $p_n=\sum_{i=1}^u (\phi a_{i})(\phi b_{i})$ with $\phi a_{i},\phi b_i\in\mathfrak{m}_{c_n}$, which is impossible by the way we chose $p_n$.
Back to the lemma: Given  $f\in K[x_1,\dots,x_c]$,  let $D_if$ denote its (formal) derivative relative to $x_i$. The lemma follows from the following general proposition.

Proposition. Let $f\in K[[x_1,\dots,x_c]]$ be a  homogeneous  polynomial of degree $n$, and let $r\in\mathbb{N}$ denote the minimal integer such that $f$ can written as $\sum_{i=1}^ra_ib_i$ with $a_i,b_i\in \mathfrak{m}_c$. Suppose that $D_1f,\dots,D_cf$ have no common zero beside the zero vector over the algebraic closure of $K$. Then $r\geq \frac{c}{2(n-1)}$.
Proof. Suppose otherwise, namely, that $f=\sum_{i=1}^r a_ib_i$ with $2r(n-1)<c$. Notice that $a_i,b_i$ are a priori not polynomials --- rather, they are power series. To fix that, we write each $a_i$ and $b_i$ as a sum of their homogenous components and rewrite the degree-$n$ homogeneous component of product $a_ib_i$ as a sum of 
  the relevant components of $a_i$ and $b_i$. By doing this, we see that $f$ can written as $\sum_{i=1}^{r(n-1)}a'_ib'_i$ with $a'_i,b'_i$ being homogeneous polynomials in $\mathfrak{m}_c$.
Let $V$ denote the affine subvariety of $\mathbb{A}^c_{\overline{K}}$
  determined by the $2r(n-1)$ equations $a'_1=b'_1=a'_2=b'_2=\dots=0$. It is well-known (see Harshorne's "Algebraic Geometry", p. 48) that every irreducible component of $V$ has dimension at least $c-2r(n-1)>0$. Furthermore, $V$ is nonempty because it contains the zero vector (because $a'_1,b'_1,a'_2,b'_2,\dots\in \mathfrak{m}_c$). Thus, there exists a nonzero $v\in \overline{K}^c$ annihilating $a'_1,b'_1,a'_2,b'_2,\dots$.
Now, by Leibniz's rule, we have $$D_jf=\sum_i D_j(a'_ib'_i)=\sum_i(D_ja'_i\cdot b'_i + a'_i\cdot D_j b'_i).$$
  It follows that $v$ above annihilates all the derivatives $D_1f,\dots,D_cf$, a contradiction! $\square$

If it weren't for the passage from power series to polynomials, the proof would work for non-homogenous polynomials and also give the better bound $r\geq \frac{c}{2}$. [Edit: This is in fact possible, see the comments.]
Such a bound would suffice to prove that the Cauchy series $x_1+x_2^2+x_3^3+\dots$ suggested in the question diverges in the $\mathfrak{m}$-adic topology.
A: A side remark (but strongly connected). Let $X$ be the set of variables (in the question, $X=\{x_1,x_2,\dots \}$) and $K[[X]]$ be the corresponding ring ($K$-algebra) of series. As a matter of fact, taking as $\mathfrak{m}$, the ideal of series without constant term, one can check (exercise) that the completion of the ring $K[X]$ (polynomials) w.r.t. the $\mathfrak{m}$-adic topology is $K[[X]]$ iff $X$ is finite. Otherwise, if $X$ is infinite, the $\mathfrak{m}$-adic completion of $K[X]$ is within $K[[X]]$, but smaller. It is the set of series $S=\sum_{\alpha\in \mathbb{N}^{(X)}}a_\alpha\, X^{\alpha}$ (multi index notation) such that, for all $n\in \mathbb{N}$, the series $S_n:=\sum_{|\alpha|=n} a_\alpha\, X^{\alpha}$ is a polynomial (for every multi degree $\alpha\in \mathbb{N}^{(X)}$, its total degree is $|\alpha|:=\sum_{x\in X}\alpha(x)$). This explains, in particular, why the sum of all variables $\sum_{x\in X} x$, which is a polynomial in the case when $X$ is finite, is not even in the $\mathfrak{m}$-adic completion of $K[X]$ in the case when $X$ is infinite.
