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

precarion wrote:

Wow... Your proof is fantastic! I was thinking about Riemann Integrals,
but you have provided the proof that can be used for Lebesgue Integrals as
well. There is only one problem... I need a proof that can be easily
explained to the studens of the 1st year of mathematics... On my
university the measure theory is on the 2nd year of studies, so I believe
that your proof unfortunately (I really like it!) will be too hard for
them to understand.

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.

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
.

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