Is 0 a natural number?
I think that modern definitions include zero as a natural number. But sometimes, expecially in analysis courses, it could be more convenient to exclude it.
Pros of considering 0 not to be a natural number:
generally speaking 0 is not natural at all. It is special in so many respects;
people naturally start counting from 1;
the harmonic sequence 1/n is defined for any natural number n;
the 1st number is 1;
in making limits, 0 plays a role which is symmetric to ∞, and the latter is not a natural number.
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