Test the following series for convergence:

The series diverges.

*Proof.* First, we write

Then, we know for all . (We can deduce this from the Bernoulli inequality, with . We proved the Bernoulli inequality in this exercise, Section I.4.10, Exercise #14.) Therefore, and we have

Since the series diverges we have established the divergence of the given series

I like the throwback to the Introductory section. Personally, I’d have used a simple limit comparison test to show.

But I’d still say yours wins out for effort.