I was once told that, since numerical integration out to infinity is not possible, that one method which is used to test whether an integral converges, say over ($0, \infty$), is to take a large interval, eg ($0, 10^6$). The method that the scientist spoke of was to cut the interval in two and test whether the upper half was smaller than the lower half $\int_0^{5x10^5}\dots > \int_{5x10^5}^{10^6}\dots$. Then to split the interval into four pieces and make sure that the upper intervals are smaller than the lower ones. I was wondering what the name of this test was.
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This "test" doesn't make a lot of sense, it will work only if the integrand is monotonocally decreasing, and then, it doesn't tell much. – May 23 '17 at 06:10
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The context that it was mentioned was Hankel transformations of functions which decay at infinity. The numerical integration is carried over an infinite number of oscillations which is numerically difficult. The method used here I believe justifying a cutoff on the integration but I'm unsure as to the name of the method used. – T-Ray May 23 '17 at 06:17