Kummer's Test Gives Characterizations for Convergence or Divergence of all Positive Series

The Second Raabe's Test and Other Series Tests

“…However, more work could be done to further generalize the Second Gauss's Test and Second Kummer's Test, as we suspect that there exists a more general formulation. In particular, we speculate that there exists a characterization of convergence and divergence of series based on the Second Kummer's Test, as was shown in the ordinary Kummer's Test [4].…”

“…This test allows more flexibility in developing a series test by selecting appropriate sequences for p n . While the statement of the test is sufficient for convergence and divergence, Tong [4] proves that the Kummer's Test in fact provides a characterization of convergence and divergence of series.…”

Evaluating the sum of convergent positive series

“…The approach of the present paper is based on the modified version of Kummer's test given by Tong [11]. We generate the test's auxiliary sequence, and on the basis of the properties of that sequence we are able to arrived at the conclusion about the required accuracy for the estimate of the series sum.…”

More on Kummer's Test

“…(Taking mk-1, one gets Cauchy's ratio criterion, comparison with the geometric series; with mk = k one is led to Raabe's criterion, etc (see [4]). )…”

“…Kummer also gave a test for divergence (not the one quoted in [4]; that one was introduced by Dini in [1]). It seems to me that the proof contains a mistake that makes it invalid; Kummer seems to assume that the sequence mkak/ak+l-mk+l 1995] NOTES is monotone decreasing, for which behavior I can't find any reason.…”