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b - c/ (n-2)! < sum (n^3-a)/ n! < b

Source: TOT 519 1996 Autumn S A2 Tournament Of Towns

August 16, 2024
algebrainequalities

Problem Statement

(a) Prove that 3−2(n−1)!<22−22!+22−23!+...+n2−2n!<33-\frac{2}{(n-1)!} < \frac{2^2-2}{2!}+\frac{2^2-2}{3!}+...+\frac{n^2-2}{n!}<3
(b) Find some positive integers aa, bb and cc such that for any n>2n > 2, b−c(n−2)!<23−a2!+33−a3!+...+n3−an!<bb-\frac{c}{(n-2)!} < \frac{2^3-a}{2!}+\frac{3^3-a}{3!}+...+\frac{n^3-a}{n!}<b
(V Senderov, NB Vassiliev)
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