Consider the series _ n=2 ^ 1 n

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

Problem

Consider the series ∑n=2∞1n ln⁡n\textstyle\boxed{\displaystyle\sum_{n=2}^{\infty}\frac{1}{n\,\ln n}} Although its terms decrease very slowly, the series has long been used as a benchmark example in the study of infinite sums. Answer the following, providing clear reasoning and precise mathematical justification. (1) Part 1: (a) Identify an elementary benchmark series whose long-run term size you can compare to 1n ln⁡n\frac{1}{n\,\ln n}. State the general term of this benchmark series. (2) Part 2: (b) For sufficiently large nn, establish an explicit inequality that relates 1n ln⁡n\frac{1}{n\,\ln n} to the corresponding term of the benchmark series you chose in part (a). Clearly indicate the value of NN past which the inequality holds. (3) Part 3: (c) Using the inequality in part (b), explain whether the given series converges or diverges. Your explanation should cite the behavior of the benchmark series and link it to the conclusion for the series in the box.

Related theory

Browse the problem library