Tommy, my person.  If 0 < a_n < a_(n+1), then the series necessarily
diverges since the n'th term is never less than a_1 and hence is
bounded away from 0.  Now if, instead, you meant

a_0 > a_1 > a_2 > a_3 > .. > 0

then your generalizatation is trivially true, if you meant, as I
suspect, that the subscripts are all the even numbers rather than the
powers of 2 as I though when I first saw your statement in connection
with the word condenation.  Not much condensation going on here.  Just
two trivial uses of the comparison test.  Why bother to publicize such
a thing?  We can all figure out the obvious for ourselves without
effort.

As for results related to the Cauchy Condensation Theorem, and so much
more, Konrad Knopp's book whose title is "Theory and Applcation of
Infinite Series" is an absolutely wonderful and very readable
reference.