Mathematics

Mathematics

Problem

Does the series ∑n=1∞n3+7nn5\displaystyle \sum_{n=1}^{\infty} \frac{n^{3}+7n}{n^{5}} converge or diverge? Give your answer and cite the value of pp for the equivalent pp-series.
Intuitive explanation

This is about recognizing that complicated rational functions can be simplified to reveal their true behavior. Start by breaking the fraction into separate terms and reducing each one—you'll find that the highest-power terms dominate, and the series becomes equivalent to a much simpler pp-series. Once you identify which pp-series it matches, the convergence test is immediate.

Intuitive explanation

This is about recognizing that complicated rational functions can be simplified to reveal their true behavior. Start by breaking the fraction into separate terms and reducing each one—you'll find that the highest-power terms dominate, and the series becomes equivalent to a much simpler pp-series. Once you identify which pp-series it matches, the convergence test is immediate.

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