Hope you all had a Merry Christmas yesterday. The challenge du jour:
Let n be an arbitrary fixed positive integer, and consider the right-hand Riemann sums of the function x^n obtained from a partition of [0, 1] into m equal intervals. These are:
[k=1, m]∑(1/m * (k/m)^n)
Using the fundamental theorem of calculus, it is of course trivial to prove that:
[m→∞]lim [k=1, m]∑(1/m * (k/m)^n) = [0, 1]∫x^n dx = x^(n+1)/(n+1) |[0, 1] = 1/(n+1).
Your mission, should you decide to accept it, is to prove the limit [m→∞]lim [k=1, m]∑(1/m * (k/m)^n) = 1/(n+1) _directly_, without using the fundamental theorem. Can you do it?