Sunday, February 28, 2016

calculus - Convergence of suminftyn=1logleft(frac(2n)2(2n+1)(2n1)right)


I have to show that the series n=1log((2n)2(2n+1)(2n1)) 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



$0<\log (1+x)0 . Therefore0<\log (\frac{4 n^2}{4 n^2-1})=\log (1+\frac{1}{4 n^2-1})< \frac{1}{4 n^2-1} < \frac{1}{2 n^2}. Thesum\sum (1/2 n^2)$ converges by Cauchy Condensation. Your sum therefore converges by Comparison.


(Where did that term 12n2 come from in the inequality? From the idea, with 4n2=x, that 1x1<2x if x is big enough.)


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