Testing for Divergence using the Integral Test

[NotGauss]

Hello all,
I was working on some homework regarding testing for convergence and divergence of series and I was having trouble with a particular series (doesn't really matter which one) and tried almost all the methods; then tried the Integral Test, my series met the conditions of the Integral Test and I integrated using integration by parts. Once I started solving the integral, I usually leave the uv terms for last since that is the simple part, I realized that I had u and v terms (using this form ∫udv = uv - ∫vdu) whos limit went to infinity.
My understanding of the Integral Test is that if the resulting integral is divergent (not a finite sum) then the summation of the series is divergent as well. So, is it enough to show that uv approaches infinity to show that my original integral (∫udv) is divergent? Or do I need to solve the improper integral of ∫vdu as well?

Thanks for your time and help!

Edit: If so, i just thought that this would be an easy way to show divergence, as you wouldn't have to fully calculate the integral but merely calculate the u and v terms.

 

[mathman]

You need to compute $\int udv$ as well, since if it is infinite it might cancel $uv$.