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Integrand of recurrent type (1+x)^r.ln^s x

Source: Romanian District Olympiad 2002, Grade XII, Problem 3

October 7, 2018
logarithmscalculusintegrationreal analysis

Problem Statement

a) Calculate lim⁡n→∞∫0αln⁡(1+x+x2+⋯+xn−1)dx, \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , for all α∈(0,1). \alpha\in (0,1) .
b) Calculate lim⁡n→∞∫01ln⁡(1+x+x2+⋯+xn−1)dx. \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx .
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