$\mathbb{Z}^{+}$ includes zero or not?

A number $x$ is defined to be positive if $x > 0$. Is $0 > 0$? No, so it is non-positive (and it is also non-negative).

$\mathbb Z^+$ is a notation, so it is difficult to argue about it, because some authors do use non-standard notation and it's alright as long as they're consistent. But $\mathbb Z^+_0$ is a better notation for the set of non-negative integers.

As some of the answers here show, in some languages the term "positive" may include $0$. In particular, it might be expected that authors whose native tongue is such a language may include $0$ in $\Bbb Z^+$.

If the book is consistent with this definition, then there's no real issue here. If the book suddenly becomes inconsistent with this definition (e.g. the author writes $\frac1n$ for $n\in\Bbb Z^+$) then it is likely a typo.