Mathematics

Mathematics

Problem

Consider the series ∑n=3∞1n(ln⁡n)2\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n(\ln n)^{2}}. Which of the following best describes its behavior? a) It converges because the integral ∫3∞ ⁣1x(ln⁡x)2 dx\displaystyle\int_{3}^{\infty}\!\frac{1}{x(\ln x)^{2}}\,dx equals 1/ln⁡31/\ln 3, a finite value. b) It diverges because its terms decay only slightly faster than 1/n1/n, so by comparison with the harmonic series it diverges. c) It converges conditionally but not absolutely, since ln⁡n\ln n merely grows without bound while the signs are all positive. d) It diverges because the integral comparison is valid only when ln⁡n\ln n appears in the numerator rather than the denominator.

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