For which real numbers does the series converge?

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

For which real numbers rr does the series ∑n=3∞1n (ln⁡n) [ln⁡(ln⁡n)]r\displaystyle \sum_{n=3}^{\infty} \frac{1}{n\,(\ln n)\,[\ln(\ln n)]^{r}} converge?

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