Sunday, February 21, 2016

measure theory - Computation of limnrightarrowinftyn(intinfty0frac11+x4+xnmathrmdm(x)C)



Let m(x) be the Lebesgue measure. I want to show that there exists CR such that
limnn(011+x4+xndm(x)C)


exists as a finite number and then compute the limit. Rewriting this gives:




limn0n(1C(1+x4+xn)1+x4+xn)dm(x)



And I thought I should use the Lebesgue dominated convergence theorem somehow to put the limit inside the integral (because calculating an integral of a rational function like this doesn't seem like a good idea) but I can't find any dominating function. Also I tried splitting into the intervals (0,1) and (1,) but this didn't help me either... Any help is appreciated!


Answer



Write the integral as
10dx1+x4+xn+1dx1+x4+xn.


The second integral goes to 0, while the first goes to 10dx1+x4, so C=10dx1+x4=π+2acoth(2)42.

Now that we know what C is, we need to estimate the error.




The second integral is bigger than 1xndx=1n+1, and smaller than (1+ϵ)1xndx, for any x, so the second integral contributes 1 to the limit.



The first integral less C equals 10xndx1+x4+xn,

which can be explicitly evaluated as
18(ψ(0)(n+58)ψ(0)(n+18)),

where ψ(0) is the digamma function, and is asymptotic to 1/2n, so the limit is 3/2. If the digamma does not make you happy, it is easy to see that the limit is 12n for the first integral. This is so because for any ϵ, the integral from 0 to 1ϵ decreases exponentially in n, and near 1 the estimate is easy to get, by approximating 1+x4 by 2.


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