A convergent series runs to the X axis and gets as close as you like; close enough, fast enough to take an area under the curve.

1/X2 as simplified example, sans series paraphernalia

A divergent series does not approach the X axis close enough or fast enough to converge adequately on the axis.

1/X

1 + 1/2 + 1/3 +1/4 ... is one such

In mathematics, a series (or sometimes also an integral) is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite. More precisely, a real or complex-valued series is said to converge absolutely if Absolute convergence is vitally important to the study of infinite series because on the one hand, it is strong enough that such series retain certain basic properties of finite sums - the most important ones being rearrangement of the terms and convergence of products of two infinite series - that are unfortunately not possessed by all convergent series. On the other hand absolute convergence is weak enough to occur very often in practice. Indeed, in some (though not all) branches of mathematics in which series are applied, the existence of convergent but not absolutely convergent series is little more than a curiosity. In mathematics, a series (or sometimes also an integral) is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite. More precisely, a real or complex-valued series is said to converge absolutely if Absolute convergence is vitally important to the study of infinite series because on the one hand, it is strong enough that such series retain certain basic properties of finite sums - the most important ones being rearrangement of the terms and convergence of products of two infinite series - that are unfortunately not possessed by all convergent series. On the other hand absolute convergence is weak enough to occur very often in practice. Indeed, in some (though not all) branches of mathematics in which series are applied, the existence of convergent but not absolutely convergent series is little more than a curiosity.

The absolute value of the common ratio is less than 1.

Those terms are both used to describe different kinds of infinite series. As it turns out, somewhat counter-intuitively, you can add up an infinitely long series of numbers and sometimes get a finite sum. And example of this is the sum of one over n2 where n stands for the counting numbers from 1 to infinity. It converges to a finite sum, and is therefore a convergent series. The sum of one over n is a divergent series, because the sum is infinity.