convergence and divergence

Convergence and divergence are the two fates of an infinite sum. A series converges if its running total settles toward a definite finite number — the additions stop mattering, and the total homes in on an answer. It diverges if the total never settles: it might race off to infinity, or swing back and forth and never pick a spot to rest.

Precisely, the series sum a_n converges if its sequence of partial sums S_n = a_1 + ... + a_n approaches a finite limit; otherwise it diverges. A few quick checks help. The n-th term test says if a_n does NOT go to 0, the series must diverge. The comparison test compares your series term-by-term with one you already understand. The ratio test looks at the long-run size ratio |a_(n+1)/a_n|: if that limit is below 1 the series converges, above 1 it diverges, and exactly 1 is inconclusive.

A crucial warning lives in the n-th term test: it only goes one way. Terms shrinking to 0 is necessary for convergence but not sufficient. The harmonic series 1 + 1/2 + 1/3 + ... has terms going to 0 yet diverges, while 1 + 1/4 + 1/9 + ... (the reciprocals of squares) also has terms going to 0 but converges. Vanishing terms alone never settle the question.