Quote:
Originally Posted by Rachel Lim
After thinking about this problem a bit more, I was wondering if this logic would work: as x→∞, the sum will resemble Σ1^(2x), which is just Σ1^(x), which diverges by the Geometric Series test (since it's just 1+1+1+1 infinite times). I get the same result, but I feel like I'm messing up something by applying l'Hopital's rule like that...
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This is not a valid approach. As a counterexample, consider the following:
sum((x/(x+3))^(x^2))
By your reasoning, as x goes to infinity, this sum will also resemble sum(1^(x^2))=sum(1^x), so it should also diverge. However, this series does not diverge.