Regular Series

A series is considered regular if it either converges or diverges - that is, if the limit of its sequence of partial sums exists as $n$ approaches infinity, whether that limit is finite or infinite: $$ \lim_{n \rightarrow \infty} s_n = (l, \infty) $$

A concrete example

Consider the following series:

$$ s_n = \sum_{k=1}^n 2k $$

The first few partial sums are:

$$ s_1 = 2 \\ s_2 = 2 + 4 = 6 \\ s_3 = 2 + 4 + 6 = 12 \\ s_4 = 2 + 4 + 6 + 8 = 20 \\ \vdots $$

As $n$ increases, the sum grows without bound:

$$ \lim_{n \rightarrow \infty} \sum_{k=1}^n 2k = \infty $$

Since the series diverges, it is still classified as a regular series.

And so on.