My textbook is showing me examples of discrete probability distributions, one of them is in the picture:
I learned in Calculus that the summation of the series $1/n$ where $n\to \infty$ is divergent. Yet this text says that it converges at $1$.
I also think that the example is bad but I'll stay on topic.
Your textbook isn't wrong. $N$ is a fixed number (6 in the example), so the sum is over a finite number of terms, which is perfectly ok.
Besides, the sum isn't the harmonic series... It is $$\frac{1}{N}+ \frac{1}{N}+ ... + \frac{1}{N}$$ where there are $N$ summands. So in the example $$ \frac{1}{6}+ \frac{1}{6}+ \frac{1}{6}+ \frac{1}{6}+ \frac{1}{6}+ \frac{1}{6}=1$$