Clarification of Root Test & Ratio Test

I read in a text book recently that when testing an infinite series for convergence, if the ratio test fails (r = 1), to not try and attempt using the root test, as it too will necessarily fail.

However, I think this is wrong, and I just want to confirm it here.

I was working this series: for which the ratio test fails and the root test gives implying convergence for this positive-termed series.

Maybe I did it incorrectly - so you should double check if you don't agree with me.