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Limits concerning an integral sequence

Source: SEEMOUS 2024, Problem 3

April 16, 2024
calculusintegration

Problem Statement

For every n≥1n\geq 1 define xnx_n by xn=∫01ln⁡(1+x+x2+⋯+xn)⋅ln⁡11−xdx.x_n=\int_0^1 \ln(1+x+x^2+\dots +x^n)\cdot\ln\frac{1}{1-x}\mathrm dx. a) Show that xnx_n is finite for every n≥1n\geq 1 and lim⁡n→∞xn=2\lim_{n\rightarrow\infty}x_n=2. b) Calculate lim⁡n→∞nln⁡n(2−xn)\lim_{n\rightarrow\infty}\frac{n}{\ln n}(2-x_n).
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