What is the name of this series test?

Question

Theorem: Suppose $a_1\ge a_2\ge a_3\ge\cdots\ge0$. Then, $\sum_{n=1}^{\infty}a_n$ converges iff $\sum_{k=0}^{\infty}2^ka_{2^k}=a_1+2a_2+4a_4+8a_8+\cdots$ converges.

I've taken to calling it the "Companion Test", but I'm sure there's a proper name for it.

Answer 1

It is called the Cauchy Condensation Test.