In any case....you were right in your setup.

So, now multiply out the binomial. Then arranging sums in the forms...

and then make use of the well know formulas for these sums.

Simplify and take the limit.

Note: Everyone knows that the limit definition is highly tedious, this is why they have been defined in terms of definite integrals, so that we may make the process easier. But, a knowledge of why definite integrals work (because of the riemann integral) is essential in an in depth understandind of them.