Monday, 17 February 2014

real analysis - Improper integral int1/20fracmathrmdttalvertln(t)rvertb




I'm working in this problem and I'm having some problems.




Study the convergence of this improper integral:



120dtta|ln(t)|b,a,b>0




For a<1 I've compared it with the integral
120dtta

and found that is convergent. When a=0, taking u=ln(t) and du=dtt we have:
120dtt|ln(t)|b=ln(12)duub
which is convergent for b>1 and divergent for b1 (is this correct?).
When a>1 I think that diverges, but cannot prove it. Any hint?


Answer



Set u=logt. Then t=eu, so dt=eudu. The limits become and log2, and we have
log2ube(1a)udu
Now, the integrand is bounded, so the problem is only for large u. In particular, there are now several cases to examine:





  1. a>1. The integral diverges, because eku grows faster than ub shrinks for any k>0, so the integrand does not tend to zero as u.

  2. a=1. The integral is log2ubdu, which we know converges if and only if b>1.

  3. $00,becauselog2ube(1a)udu<log2(log2)be(1a)udu=(log2)b11a.$


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