I have to show that the series ∑∞n=1log((2n)2(2n+1)(2n−1)) converges.
I have tried Ratio Test and Cauchy Condensation Test but it didn't work for me. I tried using Comparison Test but I couldn't make an appropriate inequality for it. Could you please give me some hints. Any help will be appreciated.
Answer
$0<\log (1+x)
(Where did that term 12n2 come from in the inequality? From the idea, with 4n2=x, that 1x−1<2x if x is big enough.)
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