📂Calculus

The p-Series and the p-Series Test

Definition¹

A series of the following form is called a $p$-series.

$$ \sum\limits_{n=1}^{\infty} \dfrac{1}{n^{p}} $$

Explanation

It is a generalization of the infinite sum of reciprocals of squares. The test introduced below can only be used for $p$-series, but its conditions and result are very simple and clear.

The $p$-Series Test

$\displaystyle \sum\limits_{n=1}^{\infty} \dfrac{1}{n^{p}}$ converges when $p \gt 1$ and diverges when $p \le 1$.

Proof

Let $f(x) = \dfrac{1}{x^{p}}$. Then $f(n) = \dfrac{1}{n^{p}}$, and $f$ is continuous and decreasing on $[1, \infty)$. Therefore, the integral test can be applied.

Integral Test

$$ \int_{1}^{\infty} f(x) dx \text{ is convergent} \iff \sum\limits_{n=1}^{\infty} a_{n} \text{ is convergent} $$

By the integral test, if the integral $\displaystyle \int_{1}^{\infty} \dfrac{1}{x^{p}}dx$ converges then the $p$-series also converges, and if the integral diverges then the $p$-series also diverges. Since this integral converges only when $p \gt 1$, the $p$-series converges only when $p \gt 1$.

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