So I woke up this morning and I was thinking about the infinite product −1×−1×−1×…, and what it equals. I came to the conclusion that it equals −i. Alternatively stated,
∞∏i(−1)=−i
Here's how I reached this:
∞∏i(−1)=eln(∞∏i(−1))=e∞∑iln(−1)=e∞∑iiπ=eiπ∞∑i1
Now, here's where I'm a little hesitant. I want to say that, from ζ(0)=−12, we can conclude that
eiπ∞∑i1=e−12iπ=−i.
I have been told before that the sum ∞∑i1 is not actually −12, but I'm not really sure why. It would seem that if this is the case, then my product would in fact not be −i. Though, I must say that −i sort of makes sense, because multiplying complex numbers is essentially rotating them, and so rotating by 180 every time will get you 180+180+180+... is the same as 180∗(1+1+1+...) which is (if my premise is right) 180∗(−12)=−90. −90 degrees on the complex plane turns out to be −i.
So my question is, is there a hole in my logic? I know what not accounting for ζ(0)=−12, the sum 1+1+1+... is divergent, but taking that into account, can I say with confidence that −1×−1×−1×⋯=−i?
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