Hopf algebra structure on $prod_n A^{otimes n}$ for an algebra $A$

For a finite dimensional $k$-algebra $A$, each $A^{otimes n}, n geq 0$ is a $k$-algebra ($A^0 = k $). Let $T= prod_{n geq 0} A^{otimes n}$. This is a $k$-alegbra with unit $(1,1,dots)$ and multiplication is component-wise. Let $Delta^{(n)} : A^{otimes n} to T otimes T$ be the deconcatenation map
$$ Delta^{(n)}(a_1 otimes dots otimes a_n) = sum_{i=0}^n (a_1 otimes dots otimes a_i) otimes (a_{i+1} otimes dots otimes a_n ). $$

I want to extend these $Delta^{(n)}$ to a comultiplication $Delta : T to T otimes T $. This does not seem to work in a straightforward way because if $t = ( t_0, t_1, dots ) in T, $ then $sum_n Delta^{(n)}(t_n)$ may not be a finite sum of pure tensors in $T otimes T$ (I have not shown this sum can be infinite, but suspect it can be).

Is there a way to make $T$ into a Hopf algebra so that $Delta(t) = Delta^{(n)}(t)$ when $t_i=0$ for $i ne n$? If not, is there an algebra similar to $T$ where this does work?

Is there a standard way to complete the tensor product and instead get a map $Delta$ from $T$ to the completion? Does this give rise to a genuine Hopf algebra or some generalization of Hopf alegbras?