- #1
sgd37
- 213
- 8
Homework Statement
Hi I need to regularize [itex]\sum_{r \in Z+1/2} r [/itex]
In my opinion there are two ways of going about it either re-express it as [itex] \sum_{r \in Z+1/2} r = \sum_{r =1} r - \frac{1}{2} \sum_{r =1} = \zeta (-1) - \zeta (0) = \frac{1}{6} [/itex]
or
[itex] \sum_{r \in Z+1/2} r = \frac{1}{2} \sum_{r =1} r - \sum_{r =1} r = - \frac{1}{2} \zeta (-1) = \frac{1}{24} [/itex]
I know I need the second answer however I don't see any reason why the first answer is not valid. In fact I think it more so, since the first sum goes term for term with the second, whereas in the second method the r =2 term of the first sum is canceled by the r=1 of the second thus having a staggered structure if the sum was finite. Any thoughts?