Lecture 20.4 - Analyzing the Bernstein Polynomials

In this video, we use what we found in the previous video about the maximum weights in the sum defining B_nf(x) to get a heuristic idea of which terms in the sum contribute most, and how many of those terms we need to collect together to obtain "most" of the value of B_nf(x). The basic outline of the actual proof that B_nf uniformly approximates f on [0, 1] essentially involves grouping the terms with "large weight" together and then bounding the error between f(x) and f(k/n) for those terms, while on the other hand the remaining terms with "small weight" we can bound by using the fact that the weights are small.