Probability distribution without a mean?

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I don't see how this is possible? how can you have a distrubution with no mean?
My professor says there is, but i don't get it...
 
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Exercise: find a series

a_1+a_2+...

that has a finite sum (and hence is can be made to have sum 1, and be a prob distribution on N) but where a_1+2a_2+3a_3+... is not finite (so there isn't a mean).

And there are less contrived examples too.
 
Simple example a_k=1/k^2
 

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