This article discusses a decreasing sequence (an) of nonnegative terms, there exist
subsequences (nk) of (n) such that P1k=1(nk+1 −nk)ank and P1n=1 an have the same
convergence. Then for P1n=1 an converges if and only if P1k=1 qkaqk converges for
1 < q ∈ N. Using these results, an extension of the ratio test is formed.
