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Part of 2023 CIIM

Problems(1)

Series of integrals

Source: CIIM 2023 - Problem 1

9/19/2023
Determine all the pairs of positive real numbers (a,b)(a, b) with a<ba < b such that the following series ∑k=1∞∫ab{x}kdx=∫ab{x}dx+∫ab{x}2dx+∫ab{x}3dx+⋯\sum_{k=1}^{\infty} \int_a^b\{x\}^k dx =\int_a^b\{x\} dx + \int_a^b\{x\}^2 dx + \int_a^b\{x\}^3 dx + \cdots is convergent and determine its value in function of aa and bb.
Note: {x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor denotes the fractional part of xx.
fractional partinfinite seriesIntegralcalculusintegration