Limit comparison - Comparison theorem for improper integrals(with explicit examples 1 Intro This is a complement to the comparison theorem for improper

Limit comparison - Comparison theorem for improper...

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Comparison theorem for improper integrals ( with explicit examples ) 1 Intro This is a complement to the comparison theorem for improper integrals in the textbook. The vanilla version presented in the textbook is good enough to solve some very easy examples and it becomes exponentially gory with the complexity of the integral. Fortunately it is not hard to refine the statement in the book, and turn it into a powerful tool to estimate the convergence of arbitrarily complicated integrals. 2 Comparison ( vanilla version ) In words: suppose that a continuous positive function h ( x ) is smaller than another continuous function f ( x ) in a neighborhood of + ; if the integral of the bigger function is convergent, then the integral of the smaller function is convergent; vice versa, if the smaller function has a divergent integral then the integral of the bigger function must diverge too. More precisely: Proposition 2.1. Let f ( x ) be a non negative continuous function on [ a, + ) . Let h ( x ) be another continuous function on the same interval such that f ( x ) h ( x ) for all x a . The following implications hold: Z + a f ( x ) dx < + ∞ ⇒ Z + a h ( x ) dx < + and Z + a h ( x ) dx = + ∞ ⇒ Z + a f ( x ) dx = + and an analogous statement holds for the interval ( -∞ , a ] and for negative functions. A similar statement holds in the neighborhood of a vertical asymptote. 3 Comparison ( enhanced version ) The idea that we try to implement is the following: the improper integral of a power function, an exponential, or a logarithm are very easy to study; we would like to reduce any given function f ( x ) to one of these functions or a product of them by comparison. This can be made precise with the definition of limit. Consider the example from the book: Z + 1 x + 1 x 4 - x 1

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