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    Why does zeta(1/2) converge to a negative number (-1.46...) rather than diverge to infinity?

    I understand how the zeta function can be rearranged to give the eta function, so I know how the value is obtained and that it is true, but this is what confuses me:

    If you take zeta(1), it diverges to infinity. So surely by dividing by 'root n' instead of by 'n' each time, the value would be even larger, so this series would also diverge. The only explanation I can think of is that a square root can be positive or negative. But then how do you know which terms are positive and which are negative?
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    The Riemann zeta function isn't defined only by the Dirichlet series. Rather, there is an analytic function which agrees with the series where it is defined, and for all other values we look at the analytic function rather than the series.
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    It depends which definition of zeta you're using. The series expansion of zeta(s) converges only for Re(s) > 1, however it's possible to extend the definition of the zeta function in such a way that it converges for a wider range of values and agrees with the series expansion for those values where the series converges.
 
 
 
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