Prove that the p-series diverges for p between 0 and 1 using the direct comparison test.

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Divergence of the p-series for p <= 1

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If \( 0 < p \le 1 \), show that the series \( \sum_{n=1}^{\infty} \frac{1}{n^p} = 1 + \frac{1}{2^p} + \frac{1}{3^p} + \cdots \) is divergent.

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Tests the ability to apply the Direct Comparison Test and manipulate basic inequalities involving exponents.

This is a foundational result in calculus and real analysis, establishing the behavior of the p-series for small powers.