Analyze the series .

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Analyze the series ∑n=2∞n(n2+1) ln⁡n\displaystyle\sum_{n=2}^{\infty}\dfrac{n}{(n^{2}+1)\,\ln n}. (1) For x≥2x\ge 2, consider g(x)=x(x2+1) ln⁡xg(x)=\dfrac{x}{(x^{2}+1)\,\ln x}. Verify that gg satisfies all conditions required to compare an improper integral with the given series. (2) Evaluate the improper integral ∫2∞x(x2+1) ln⁡x dx\displaystyle\int_{2}^{\infty}\dfrac{x}{(x^{2}+1)\,\ln x}\,dx. Show each step clearly, including any substitutions or integration techniques you employ. (3) Using your result from part (b), determine the convergence or divergence of the series. Justify your answer in the context of the work shown in parts (a) and (b).

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