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

e.g.

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

with T(n) = triangular number.

but probably the cases a_n = oo <=> sum ??? = oo

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