MathDB

7

Part of 1950 Miklós Schweitzer

Problems(2)

Miklos Schweitzer 1950_7

Source: first round of 1950

10/2/2008
Let x x be an arbitrary real number in (0,1) (0,1). For every positive integer k k, let fk(x) f_k(x) be the number of points mx\in [k,k \plus{} 1) m \equal{} 1,2,... Show that the sequence f1(x)f2(x)⋯fn(x)n \sqrt [n]{f_1(x)f_2(x)\cdots f_n(x)} is convergent and find its limit.
floor functionceiling functionalgebra proposedalgebra
Miklos Schweitzer 1950_7

Source: second part of 1950

10/3/2008
Examine the behavior of the expression \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n as n→∞ n\rightarrow \infty
logarithmsintegrationcalculusEulerfunctionreal analysisreal analysis unsolved