Mathematics

Mathematics

Problem

Determine whether the series ∑n=2∞ln⁡nn2\displaystyle \sum_{n=2}^{\infty} \frac{\ln n}{n^{2}} converges or diverges.
Intuitive explanation

The key is recognizing that ln⁡n\ln n grows much slower than any positive power of nn. Even though ln⁡n\ln n increases without bound, the denominator n2n^2 dominates so heavily that the terms shrink fast enough for the series to converge. Compare this series to a pp-series or use the integral test to formalize that intuition.

Intuitive explanation

The key is recognizing that ln⁡n\ln n grows much slower than any positive power of nn. Even though ln⁡n\ln n increases without bound, the denominator n2n^2 dominates so heavily that the terms shrink fast enough for the series to converge. Compare this series to a pp-series or use the integral test to formalize that intuition.

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