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Explain the convergence/divergence of a p-series.

A p-series ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^{p}} converges if p > 1 and diverges if p ≤ 1.

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Explain the convergence/divergence of a p-series.

A p-series ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^{p}} converges if p > 1 and diverges if p ≤ 1.

Explain the behavior of the harmonic series.

The harmonic series ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n} diverges, even though its terms approach zero.

Why is simplifying the terms important before applying the p-series test?

Simplification ensures the series is in the standard 1np\frac{1}{n^p} form, allowing for correct identification of 'p' and accurate convergence/divergence determination.

Define a p-series.

A series of the form ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^{p}}, where p is a constant.

Define a harmonic series.

A p-series where p = 1, represented as ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}.

What does it mean for a series to converge?

The sum of the infinite terms approaches a finite value.

What does it mean for a series to diverge?

The sum of the infinite terms does not approach a finite value; it tends to infinity or oscillates.

What is the general form of a p-series?

∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^{p}}

What is the formula for the harmonic series?

∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}