# P series

A p-series takes on the form,

,

where p is any positive real number. P-series are typically used as a test of convergence; if p > 1, the p-series converges; if 0 < p ≤ 1, the p-series diverges. This test is referred to as the p-series test, and is a corollary of the integral test.

The integral test helps determine whether a series converges or diverges by comparing it to an improper integral. More specifically, suppose that f is a continuous function that is positive and decreasing over an interval [k, ∞]. Then, the infinite series,

converges if and only if the integral,

is finite. If the integral diverges, then the series also diverges.

Let , which is a positive, continuous, decreasing function when x ≥ 1. Since for all x = n, we can use the integral test to check the convergence of . For the case where p > 1:

Since p > 1, p - 1 > 0, and b^{p-1} → ∞ as b → ∞, so:

Since p > 1, is some positive real number, so converges to . Thus, also converges by the integral test.

For the case 0 < p ≤ 1, if 0 < p < 1, p - 1 < 0, so 1 - p > 0. The limit of the integral then becomes:

Since diverges, also diverges by the integral test.

For the case where p = 1,

Thus, also diverges by the integral test.

Note that the sum of the series is not equal to the value of the integral. The integral test only tells us that the series converges if the integral converges.

Examples

i. The expression can also be written as . Since p = ⅓ < 1, the p-series diverges by the p-series test.

ii. The expression can also be written as . Since , the p-series converges by the p-series test.

ii. The expression can also be written as . Since , the p-series converges by the p-series test.