Compute .

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- Feb 25th 2011, 03:38 PMchiph588@Limit and integral
Compute .

- Feb 27th 2011, 02:31 PMgirdav
Let . The sequence is decreasing and positive so the limit exists. We have

and by the substitution we get .

For all we have the inequality hence .

Now we try to compute . By putting we get

. We can see that the two integrals are the same (by the substitution ). So we only have to compute the first one.

We have . Then and we conclude the limit we were looking for is . - Feb 27th 2011, 07:43 PMBruno J.
It is an immediate consequence of the dominated convergence theorem that the limit is 0.

- Feb 27th 2011, 07:53 PMRandom Variable
- Feb 27th 2011, 08:14 PMBruno J.
The integrand is non-negative and converges monotonically to almost everywhere, so you may switch the limit and the integral.