Category:Definitions/Properly Divergent Series

This category contains definitions related to Properly Divergent Series.
Related results can be found in Category:Properly Divergent Series.

Let $s$ be an infinite series:

$s = \ds \sum_{k \mathop = 1}^\infty a_k = a_1 + a_2 + a_3 + \cdots$

Then $s$ is a properly divergent series if and only if it is a divergent series such that either:

$s_n \to \infty$ as $n \to \infty$

or:

$s_n \to -\infty$ as $n \to \infty$

where $s_n$ is the $n$th partial sum of $s$:

$s_n = \ds \sum_{k \mathop = 1}^n a_k$