> > The purpose of this post is to archive an old
> > handout
> > of mine on the Cauchy condensation test. The handout
> > was
> > originally written in January 1998 (I believe I had
> > earlier
> > prepared a handwritten version, in Jan. or Feb.
> > 1997)
> > and the audience consisted of the students in a high
> > school calculus 3 course that I taught several times
> > in
> > the late 1990s. The course covered (at a level
> > somewhat
> > higher than in standard college level calculus
> > courses)
> > improper integrals, sequences and series, parametric
> > equations, some complex variables and other
> > miscellaneous
> > topics (e.g. an introduction to Fourier series
> > expansions),
> > partial differentiation, and the beginnings of
> > multiple
> > integration.

> > Dave L. Renfro

> generalizations might be intresting.

> tommy condensation test :)

> 0 < a_n < a_n+1

> a0 + a1 + a2 + a3 + ... < oo <=> 2 a0 + 2 a2 + 2 a4 + ... < oo

> x)

> in general for rising g(n) E N and g'(n) > 1

> sum g(n)a_g(n) < oo <-> sum a_n < oo

> sum T(n)a_T(n) < oo <-> sum a_n < oo

> with T(n) = triangular number.

> are the intresting ones.

> im convinced earlier work has been done on such matters.

> btw

> (under trivial conditions)
> you can use zeta-expansion of a series :

> sum f(n) = sum g(1/n) = sum a0 + a1 1/n + a2 1/n^2 + ...

> = a0 + a1 zeta(1) + a2 zeta(2) + ...

> and prove almost the entire case ( again under trivial conditions ) with the
> harmonic sum.

> but that is more complicated then neccassary.

> also , i suppose you ( dave l renfro ) is already aware of the above.

> and thus in a sense my post is more aimed at others.
> (not at musatov)

> i suspect the handout occured during taylor series radius of convergeance
> lectures because of the geometric trend.

> i think there might be potential in applying multiple combinations of
> condensation and anti-condensation to find properties - closed forms -
> boundaries of double/triple series or double/triple products.

> is there a non-trivial relation to special functions worth mentioning ?

> regards

> tommy1729