Is infinite product of nilpotent (solvable) groups nilpotent (solvable)?
(I known they are true for finite cases)
Is infinite product of nilpotent (solvable) groups nilpotent (solvable)?
(I known they are true for finite cases)
No to both. Take a product of groups $G_n$ which are $n$-step nilpotent (resp. $n$-step solvable) but not $n-1$-step nilpotent (resp. $n-1$-step solvable). However, this is true if (and only if!) the nilpotence (resp. solvability) degree is uniformly bounded.
I don't have enough rep. to comment the following remark.
Each term of the lower central series (resp. derived series) commutes with an arbitrary cartesian product.
This also proves the last assertion of Qiaochu.
I think a correct argument is the following.
The cartesian product $\prod G_i$ satisfies a law $w$ if and only if each $G_i$ satisfies $w$.
Nilpotent groups of class at most $c$ are precisely the groups satisfying the law $[x_1,\ldots,x_{c+1}]=1$.