Existence of Improper integral

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Main Question or Discussion Point

Let A be a constant.

Let f(t) be an integrable function in any interval.

Let h(t) be defined on [0, oo[ such that
h(0) = 0
and for any other "t", h(t) = (1 - cos(At)) / t

It is not difficult to see that h is integrable on [0, b] for any positive "b", so fh is also integrable in said interval.

Considering fh as an integrand, Let the function G(b) be defined in the domain [0, oo] and = to the definite integral from 0 to b.

How to proof that lim G(b) exists? (b --> oo)

(In pg. 472 of first edition of Mathematical Analysis Apostol says that it indeed does so).

Sorry for not using latex but there is some technical problem in some server...
 

Answers and Replies

240
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I must make an amend.

For defining the function G, take the absolute value of said integrand.

This implies that G is an monotonic increasing function.

So, all that we have to prove is that G is a bounded function. How to do it? Ideas?
 
mathman
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By taking the abs. value of the integrand, you have made G(b) divergent - for large t, the integrand behaves like 1/t. If you don't take abs. value, I suspect it will converge - similar to alternate sign harmonic series.
 
240
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I'll try that. Thank you.
 

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