# 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

- Definition:General Harmonic Numbers: where the index remains finite

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

## Sources

- 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*(2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests