$p$ -series¶

Definition¶

The $p$ -series is defined as the infinite series $\displaystyle\sum _{n=1}^{\infty }\frac {1}{n^{p}}\text{,}$ for any real number $p\text{.}$

Interpretation¶

The $p$ -series are the reference series for terms that decay like a power of $n$ . For $p>0$ the terms $\frac{1}{n^p}$ tend to $0$ , and $p$ measures how fast; the series converges if and only if $p>1$ $1$ , so the harmonic series, with $p=1$ $1$ , sits exactly on the boundary.

Partial sums of the series of 1 over n to the p for p = 2, 1 and 1/2; the first levels off at pi squared over 6, while the other two keep growing.

For $p=2$ (blue) the partial sums level off below the dashed line at $\frac{\pi^2}{6}\approx1.645$ $1.645$ ; after $20$ terms they are about $1.596$ . For $p=1$ (red) and $p=\frac{1}{2}$ (orange) the partial sums have reached about $3.60$ and $7.60$ and keep growing. The red sums grow only like $\ln n$ , so slowly that the picture alone cannot show divergence; that needs a proof, by grouping terms or by the integral test.

Lower $p$ towards $1$ : the limit grows and the partial sums take longer to reach it. At $p\le 1$ they keep growing, even though slowly.

The summands come from the power function $x^{-p}$ at positive integers. For $p>0$ $0$ , the integral test compares them with an improper integral; its $p$ -test gives convergence for $p>1$ $1$ . For $p\le0$ $0$ , the terms do not tend to zero, so the series diverges.

The $p$ -series are the usual benchmarks for the comparison test for series: a series whose terms behave like $\frac{1}{n^p}$ for large $n$ converges exactly when $p>1$ $1$ . A common slip is to confuse them with the geometric series. In $\displaystyle \sum\frac{1}{n^p}$ the exponent is fixed and the base $n$ varies, while in $\displaystyle \sum q^n$ the base is fixed: $\displaystyle \sum\left(\tfrac{1}{2}\right)^n$ converges, but $\displaystyle \sum n^{-1/2}$ diverges.

Key Results¶

The series converges if and only if $p>1\text{.}$
The series diverges for $p\le 1\text{.}$
The convergence behaviour depends entirely on the exponent.

Examples¶

$\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$ converges.
$\displaystyle \sum_{n=1}^\infty \frac{1}{n}$ diverges.
$\displaystyle \sum_{n=1}^\infty \frac{1}{\sqrt{n}}$ diverges.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 12

Aliases: p series, p-series, hyperharmonic series Symbols: infinity, pi, square root, sum

Last updated: 20 October 2026 03:02

ppp-series¶