This is outside my area, but it looks like an application for L'Hopital's Rule.
Something to try unless someone else speaks up!
I mean, friend. It's Riemann sums. Plain and simple. You can do a laborious argument to show that it's bounded above and below by terms that approach one-tenth.
Or, you can just note that this is . This sum can be proved by induction to be . Take the limit and you will get one-tenth. Ta-da!
For the edification of the OP: Power Sum -- from Wolfram MathWorld