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I have had absolutely no sleep for a while, so my math brain has been failing me. Last night, I was working on a problem and I believe that one of my connections is wrong. Is this number countably infinite or uncountably infinite (asmapproaches infinity):

[itex]\sum_{n=1}^{m}{(\frac{m!}{n!})^{a}}[/itex]

I feel like it is countably infinite, but I am not sure.

Also, what about just simply (m!)^{a}as m approaches infinity? If it is uncountable, that will be rather cool, too, as that is what one of my proofs hinge on.

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# How big is this number?

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