Abstract
SummaryGiven a decreasing sequence of nonnegative terms, we prove that there exist subsequences of {n} such that and have the same convergence. In particular, converges if and only if 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.
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