Defining Convergent and Divergent Infinite Series

You can't add infinitely many numbers one at a time and ever finish, so convergence is defined through a shortcut: build the sequence of partial sums, S1, S2, S3, and so on, where S_n is the sum of the first n terms. Then ask what that sequence of sums does as n grows without bound. If the partial sums home in on one specific number L, the series converges and its sum is defined to be L. If the partial sums grow without bound, bounce around forever, or drift off in some other way without settling, the series diverges — it simply has no sum.

This reframes an impossible task, adding forever, into a familiar one: finding the limit of a sequence. That's the whole mechanism. Nothing about the individual terms guarantees convergence by itself; what matters is whether the running totals stabilize. A series can have terms shrinking toward zero and still diverge, or have terms that look unruly yet produce partial sums that settle down. The partial sum sequence is the only honest judge.

This definition is the foundation for everything else in the unit. Once you can picture partial sums geometrically, the geometric series formula becomes an evaluation of that same limit, and the convergence tests that come next (integral, comparison, ratio, alternating) are really just shortcuts for predicting whether the partial sums will settle without having to compute them all. Keep this picture of a sequence of running totals in mind, because every test later is just a faster way to answer the same question: where, if anywhere, does that sequence go?