# Definition:P-Series

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## Theorem

Let $p \in \C$ be a complex number.

The series defined as:

$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^p} = 1 + \frac 1 {2^p} + \frac 1 {3^p} + \frac 1 {4^p} + \dotsb$

is known as a $p$-series.

## Also defined as

Authors whose focus is on real analysis define a $p$-series for real $p$ only.

## Also known as

Some sources dispose of the hyphen: $p$ series.

## Also see

• Results about $p$-series can be found here.