the sum, , was chosen so that all terms of the series are positive. Notably, these series provide examples of infinite sums that converge or diverge arbitrarily slowly. For instance, in the case of and , the partial sum exceeds 10 only after (a googolplex) terms; yet the series diverges nevertheless. Schlömilch's generalization A generalization of the condensation test was given by Oskar Schlomilch. Let be a strictly increasing sequence of positive integers such that the ratio of successive differences is bounded: there is a positive real number , for which Then, provided that meets the same preconditions as in Cauchy's convergence test, the convergence of the series is equivalent to the convergence of Taking so that , the Cauchy condensation test emerges as a special case. References Bonar, Khoury (2006). Real Infinite Series. Mathematical Association of America. . External links Cauchy condensation test proof Augustin-Louis Cauchy Convergence tests