# Sequences of Integrals

1. Jul 29, 2009

### eok20

I sometimes have difficulty knowing how to approach problems where you have to evaluate a limit of a sequence of (Riemmann) integrals. I know that when the functions converge uniformly you can bring the limit inside. But when there is not uniform convergence, I was wondering if there are any other theorems/tricks that work.

For example, say you want to show that the integral of (sin x)^n over any finite interval [a,b] goes to 0 as n goes to infinity. Now (sin x)^n converges pointwise to 0 whenever x is not k*pi, for some integer k. So my thought is it split up the interval like this: [a, k*pi - eps/2], [k*pi - eps/2, k*pi + eps/2], [k*pi + eps/2, b] (and of course do this for every integral multiply of pi in the interval). Now on the sets like [k*pi + eps/2, b], sin x is bounded by a number less than 1 and reaches that bound (since the set is compact) and so on that interval (sin x)^n goes to 0 uniformly. On the sets like [k*pi - eps/2, k*pi + eps/2], |(sin x)^n| <= 1 so the absolute value of the integral is less than eps. Therefore the whole integral converges to 0. Is this a good and typical way of attacking a problem like this? Are there other ways to do these?

Thanks!