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Not true. Infinite sums are not defined for series that don’t converge, and sum(1..n) does not converge as n => infinity.

Yes, you can play a nice trick with that sum if you ignore the fact that inf-inf is meaningless.

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It depends on how you define the summation of infinite series. With the standard convergence type definitions you are correct. But it is possible to define this sum in a consistent way (e.g. Ramanujan summation or Riemann Zeta analytic continuation) as shown in Hardy's “Divergent Series”. The cost is that they have properties like rearranging the order of the terms gives a different result. Apparently this sum can come up in Quantum Field Theory when calculating vacuum force between two conducting plates.

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