Discuss the convergence or divergence of the series .
The terms of this series go to 0, but does it converge? Using the Cauchy condensation test, it is true that this series and
converge or diverge together. It is well known that the terms of increase much more quickly than the terms of so we know that these terms do not go to 0, but instead increase to and thus this series diverges. We can prove this using the ratio test:
Since this number is greater than 1, by the ratio test, this series diverges. Since this series diverges, by the Cauchy condensation test, our original series also diverges.