(a) For the series , the ratio test gives the limit , and the direct comparison test…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Accumulation: integration and the techniques that make it possible

Problem

(a) For the series ∑n=2∞1nln⁡n\displaystyle\sum_{n=2}^{\infty}\frac{1}{n\ln n}, the ratio test gives the limit 11, and the direct comparison test against ∑1/n\sum 1/n or ∑1/n2\sum 1/n^2 settles nothing either way. Use the integral test to decide whether this series converges or diverges. State the bracketing inequality you use and evaluate the relevant improper integral. (b) A second calculation from the same family is ∫1eln⁡xx dx\displaystyle\int_1^{e}\frac{\ln x}{x}\,\mathrm{d}x. Evaluate it, then check the result by differentiating your antiderivative back with the derivative rules of Chapter 1.

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