TITLE: A generalization of Cauchy's condensation test
QUESTION [6 upvotes]: If $(a_n)$ is a decreasing sequence of positive real numbers and $(u_n)$ is a strictly increasing sequence 
of positive integers such that $ \dfrac {u_{n+2}-u_{n+1}}{u_{n+1}-u_n}$ is bounded , then how do we prove that the 
convergence of $\sum_{n=1}^ \infty (u_{n+1}-u_n)a_{u_n}$ implies the convergence of $\sum_{n=1}^ \infty a_n$ ?

REPLY [5 votes]: Suppose that
$$
\frac{u_{n+2}-u_{n+1}}{u_{n+1}-u_n}\le M\tag{1}
$$
Simply counting terms and using the monotonicity of $a_k$, we get
$$
(u_{n+1}-u_n)a_{u_{n+1}}\le\sum_{u_n\le k\lt u_{n+1}}a_k\le(u_{n+1}-u_n)a_{u_n}\tag{2}
$$
summing $(2)$ yields
$$
\sum_{n=1}^\infty(u_{n}-u_{n-1})a_{u_n}\le\sum_{k=1}^\infty a_k\le\sum_{n=0}^\infty(u_{n+1}-u_n)a_{u_n}\tag{3}
$$
Applying $(1)$ gives
$$
\frac1M\sum_{n=1}^\infty(u_{n+1}-u_{n})a_{u_n}\le\sum_{k=1}^\infty a_k\le\sum_{n=0}^\infty(u_{n+1}-u_n)a_{u_n}\tag{4}
$$