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