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Looks to me like there are errors in the transformation. The sum term corresponding to the resulting equation of the proof has a wrong sign in the second summand of the left term (+ instead of -). Also in the original manuscript the right side always starts with a positive alternating series term, which is correct. In the version on the margin the sign of the first term of the right side depends on the evenness of a.

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I see your point. There are indeed a couple of errors in there. But the point still holds.

The sign is wrong on the sum of the left hand side. And a (-1)^(a+1) is missing on the RHS. But these do not change the "this is an integer" claim.

Further down, it does assume that 'a' is even, without saying so . But if 'a' was odd, things would still be the same but we would have a ]-1,0[ interval.

Anyway, I'm not a mathematician so I'm probably missing stuff.

Thank you for confirming this. These kind of errors drive me crazy when I try to work through proofs on my own.

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