The Reimann "Zeta" function: How can it ever converge?

So I was reading in Wikipedia about the Zeta function, which is defined as:

    Z(s) = 1/(1**s) + 1/(2**s) + 1/(3**s) + ...

Both the domain and range are specified as the complex numbers.

And it says that if s is a negative integers (-2, -4, -6, etc), then Z(s)
is zero.  But that can't be right.  But first, a little manipulation:

Suppose s is -2:

    1/(n**s), where s = -2

is:

    1/(1/(n**2))

is:

    n**2

so, the sum is like:

    1+4+9+16+25+...

Which just grows without bounds.  And is certainly never zero.

So, is Wikipedia wrong?  Or just a typo?

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