MathDB

Problem 3

Part of 2000 VJIMC

Problems(2)

limits equal for bounded sequence of reals

Source: VJIMC 2000 1.3

7/26/2021
Let a1,a2,…a_1,a_2,\ldots be a bounded sequence of reals. Is it true that the fact lim⁡N→∞1N∑n=1Nan=b and lim⁡N→∞1log⁡N∑n=1Nann=c\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=cimplies b=cb=c?
limitsreal analysis
Czech Republic 2000

Source:

6/6/2015
Prove that if m,n are nonnegative integers and 0<=x<=1 then (1−xn)m+(1−(1−x)m)n≥1(1-x^n)^m + (1-(1-x)^m)^n \ge 1
inequalitiesfunction