Two Convergence Theorems and an Extension of the Ratio Test for a Series

Summary: Given a decreasing sequence (Formula presented.) of nonnegative terms, we prove that there exist subsequences (Formula presented.) of {n} such that (Formula presented.) and (Formula presented.) have the same convergence. In particular, (Formula presented.) converges if and only if (Formula presented.) converges for any natural number q > 1. Using this result, the ratio test is also extended. One example further shows that the new extended ratio test is applicable to discussing the convergence of a p-series for which the ratio test fails.