Does the Limit of G(b) Exist as b Approaches Infinity?

[Castilla]

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...

 

[Castilla]

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]

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.

 

[Castilla]

I'll try that. Thank you.