# 2014-02 Series

Determine all positive integers $$\ell$$ such that $\sum_{n=1}^\infty \frac{n^3}{(n+1)(n+2)(n+3)\cdots (n+\ell)}$ converges and if it converges, then compute its value.

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Let $$f$$ be a real-valued continuous function on $$[ 0, 1]$$. For a positive integer $$n$$, define
$B_n(f; x) = \sum_{j=0}^n f( \frac{j}{n}) {n \choose j} x^j (1-x)^{n-j}.$
Prove that $$B_n (f; x)$$ converges to $$f$$ uniformly on $$[0, 1 ]$$ as $$n \to \infty$$.