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Ask question asked 12 years, 5 months ago modified 1 year, 2 months ago How to determine whether an improper integral converges or diverges My question is how can i estimate the value of an improper integral from $[0,\\infty)$ if i only have a programming routine that gives me the function evaluated at 100 data points, or 100 values of.

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I know that improper integrals are very common in probability and statistics

Also, the laplace transform, the fourier transform and many special functions like beta and gamma are defined using improper integrals, which appear in a lot of problems and computations

But what about their direct, practical applications in real life situations? Improper integrals can be defined as limits of riemann integrals All you need is local integrability However, we know that continuity is almost necessary to integrate in the sense of riemann, so teachers do not worry too much about the minimal assumptions under which the theory can be taught.

My calculus professor mentioned the other day that whenever we separate an improper integral into smaller integrals, the improper integral is convergent iff the two parts of the integral are conver. What is the general way of determining whether you should use direct comparison vs limit comparison for finding if improper integrals are convergent or divergent I normally look at the solutions and i'm able to understand what they are doing but i don't understand the thought process of choosing a specific test. What is the difference between improper integrals and the a series

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For example, if you solve a type one improper integral from 1 to infinity, the answer is different than if you solve the same fun.

The improper integral $\int_a^\infty f (x) \, dx$ is called convergent if the corresponding limit exists and divergent if the limit does not exist While i can understand this intuitively, i have an issue with saying that the mathematical object we defined as improper integrals is convergent or divergent. Hartman and mikusinski's book the theory of lebesgue measure and integration make an interesting remark on improper integrals in multiple dimensions In the case of one variable, we introduced, besides the concept of the lebesgue integral on an infinite interval, the further concept of an improper integral.