A series being convergent means it sequence of partial sums is convergent. A sequence is convergent iff it's Cauchy, meaning for every epsilon > 0 blah blah blah.
>A sequence <math display="block">x_1, x_2, x_3, \ldots</math> of real numbers is called a Cauchy sequence if for every [[Positive and negative numbers|positive]] real number <math>\varepsilon,</math> there is a positive [[integer]] N such that for all [[natural numbers]] <math>m, n > N,</math> <math display="block">|x_m - x_n| < \varepsilon,</math> where the vertical bars denote the [[absolute value]].
If it's not convergent, there is an epsilon > 0 which is a counterexample. The "blah blah blah" is false for epsilon = 0. We can use the supremum of the epsilons for which "blah blah blah" is false to measure how divergent a sequence is (if the epsilons are not bounded, assign infinity). So we have a scale of divergence from 0 to infinity inclusive where 0 is convergent and anything above that is divergent. On this scale all the convergent stuff would be on the boundary.
Now I wonder if we can we do it the other way around, finding a number that measures how convergent a sequence is.
Equivalently we can use limit superior - limit inferior, taking the ∞ - ∞ and (-∞) - (-∞) cases to be ∞.
always seemed odd to me that series like 1-1+1-1+1-1+1-1+1-1+... were called divergent
My first thought was that you meant anime shows by series, and divergent or convergent in a sense of entertaining or tense.