# 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.

GD Star Rating
loading...

# Solution: 2014-01 Uniform convergence

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 $$n \to \infty$$.

The best solution was submitted by 김범수. Congratulations!

Similar solutions are submitted by 권현우(+3), 박경호(+3), 오동우(+3), 이시우(+3), 이종원(+3), 이주호(+3), 장경석(+3), 장기정(+3), 정성진(+3), 정진야(+3), 조준영(+3), 채석주(+3), 한대진(+3), 황성호(+3). Thank you for your participation.

GD Star Rating
loading...