Monday, 25 April 2016

integration - Find limnrightarrowinftyfrac1nintinfty1fracmathrmdxx2log(1+fracxn)



Find:



limn1n1dxx2log(1+xn)



The sequence 1nx2log(1+xn)=1x3log(1+xn)xn converges pointwise to 1x3. So if we could apply Lebesgue's Dominated Convergence Theorem, we have:




limn1n1dxx2log(1+xn)=limn1dxx3=12



I have a problem with finding a majorant. Could someone give me a hint?


Answer



I showed in THIS ANSWER, using only Bernoulli's Inequality the sequence (1+xn)n is monotonically increasing for x>n.



Then, we can see that for x1 and n1, the sequence fn(x) given by



fn(x)=nlog(1+xn)




is also monotonically increasing. Therefore, a suitable dominating function is provided simply by the inequality



1nlog(1+xn)1log(1+x)1log(2)



Therefore, we have



1nx2log(1+xn)1x2log(2)



Using the dominated convergence theorem, we can assert that




limn11nx2log(1+xn)dx=1limn(1nx2log(1+xn))dx=11x3dx=12.


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