The series diverges if

Usable when is a continuous, positive, decreasing function on the interval

Applies as long as two series , are both positive:

Also applies as long as long as the two series are positive, but only if exists, is positive and is finite. If these conditions are met, converges only if converges.

If converges, then also converges. This is called absolute convergence, and we say that converges absolutely.

We study the following limit

We study another limit

We use the alternating series test when we have some form of or
Such series converge if and only if and