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The fraction has a limit - [Iran Second Round 1988]

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December 7, 2010
algebrapolynomialinductionfactorialalgebra proposed

Problem Statement

(a) Prove that for all positive integers m,nm,n we have ∑k=1nk(k+1)(k+2)⋯(k+m−1)=n(n+1)(n+2)⋯(n+m)m+1\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}
(b) Let P(x)P(x) be a polynomial with rational coefficients and degree m.m. If nn tends to infinity, then prove that ∑k=1nP(k)nm+1\frac{\sum_{k=1}^n P(k)}{n^{m+1}} Has a limit.
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