I'm trying to prove (or disprove) a statement by Feld:
https://math.feld.cvut.cz/mt/txte/2/txe3ea2d.htm
It's a generalization of Raabe's test and it says that for a sequence $a_n$, for all n, if $a_n>0$ and $c_n$ is absolutely convergent and
$a_{n+1}/a_n = 1-A/n+c_n$
Then $\sum_{n=1}^\infty a_n$ converges iff $A>1$
I rewrite:
$n(1-a_{n+1}/a_n)=A-n c_n$
and take the limit on both sides as $n \to \infty$ and its proven by Raabe's simple test, as long as $\lim_{n \to \infty} n c_n=0$.
But it has been shown (Series converges implies $\lim{n a_n} = 0$) that $\lim_{n \to \infty} n c_n$ is not zero unless $c_n$ is non-increasing, so I'm stuck. Did Feld forget to add that $c_n$ is non-increasing, or is he right as it stands?
