Cauchy condensation test The following is Cauchy condensation test:
If $b_n \ge 0$ and $b_n \ge b_{n+1}$ then $\sum b_n$ converges if and only if $\sum 2^n b_{2^n}$ converges.
Is the following modified Cauchy condensation test possible:
Let $N \in \mathbb N$. Then if $b_n \ge 0$ and $b_n \ge b_{n+1}$ then $\sum b_n$ converges if and only if $\sum N^n b_{N^n}$ converges?

There is an even stronger theorem due to Schlömilch, see Cauchy condensation test. In the notation of this web site take $u_n=N^n$. Then $\Delta u_n/\Delta u_{n-1}=N$ which is bounded and $\Delta u_n=(m-1)m^n$. Hence, your generalization is a special case of Schlömlich's Theorem.
