Convergence of P-Series

Theorem

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

Absolute Convergence if $\map \Re p > 1$

Let $\map \Re p > 1$.

Then the $p$-series:

$\ds \sum_{n \mathop = 1}^\infty n^{-p}$

converges absolutely.

Divergence if $0 <\map \Re p \le 1$

Let $0 < \map \Re p \le 1$.

Then the $p$-series:

$\ds \sum_{n \mathop = 1}^\infty n^{-p}$

diverges.

Real Case

Let $p \in \R$ be a real number.

Then the $p$-series:

$\ds \sum_{n \mathop = 1}^\infty n^{-p}$

is convergent if and only if $p > 1$.

Also see

The mapping $\ds p \mapsto \sum_{n \mathop = 1}^\infty n^{-p}$ is well-known as the Riemann zeta function.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): $p$-series
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): $p$-series