Daily Archives: March 14, 2014

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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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.

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