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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 $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$ converges if p > 1 and diverges if p ≤ 1.
Explain the behavior of the harmonic series.
The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ 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 $\frac{1}{n^p}$ form, allowing for correct identification of 'p' and accurate convergence/divergence determination.
What is the general form of a p-series?
$\sum_{n=1}^{\infty} \frac{1}{n^{p}}$
What is the formula for the harmonic series?
$\sum_{n=1}^{\infty} \frac{1}{n}$
Define a p-series.
A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$, where p is a constant.
Define a harmonic series.
A p-series where p = 1, represented as $\sum_{n=1}^{\infty} \frac{1}{n}$.
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.