Re: Proof of Dirichlet's Test for convergence of given integral

On Fri, 08 Jun 2007 13:43:08 +0100, Timothy Murphy
<tim@xxxxxxxxxxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

Do you possibly know any proof of Dirichlet's Test for convergence of
integrals that is using only Riemann Integrals? (I've already found one
in Fichtenholtz's calculus book, but it's too boring in my opinion, and
I'm still looking for something else...)

I would have thought the simplest way would be to first prove
the corresponding result for series sum a_n b_n ,
which is well-known and useful in many cases
(eg in studying the convergence of Dirichlet series sum a_n n^{-s}).

If in fact the integrals involved were all Riemann integrals
then the series result would extend at once to the integral result,
using the usual approximation to the integrals by sums.

Is this entirely clear? It may well be so, and in fact it may
well be trivial, but it's not entirely clear to me that it's
going to work with no problem (not that I've tried writing
it down carefully).

What bothers me is that an _improper_ Riemann integral
is not a priori approximated by a Riemann sum, it's by
definition the limit of integrals over compact intervals,
and _those_ integrals are approximated by Riemann sums.
So it seems like there's going to be an interchange of
limits at some point, and if swapping two limiting
operations always worked with no problem most theorems
of analysis would be trivial.

Seems possible to me that it works, but that it may
not work just by applying the result for sums per se,
rather one is going to need to insert the proof of the
result for sums and keep track of the epsilons. (???)

I would have thought it was easy enough ...

To show I(a,infty) converges, one has to show that
I(a,b)->0 as a,b->infty, ie given eps there is a C such that

|I(a,b)| < eps if a,b > C.

So really one is dealing with a proper integral.

Since g(x)->0 monotonically,

g(x) < eps if x > C.

(I assume for simplicity that g(x) >= 0.)

Also by hypothesis,

|I(a,b)| < D.

Then it follows from the re-arrangement of series
(and approximation of the integral by sums) that

| int_a^b f(x) g(x) dx | <= 3C eps.

I don't quite follow all the details there (maybe
there were some typos?), but yes, some argument
more or less like that should be fine. That argument
is exactly as I suggested: It's not using the theorem
on _convergence_ of sums per se, it's inserting the
proof of that theorem and keeping track of the
epsilons.


************************

David C. Ullrich
.

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