Partial Sums and Convergence

Real Analysis · Axiom Academy

LESSON Partial Sums and Convergence A series means nothing until you read its sequence of partial sums S_n — and that sequence decides everything. 1. A Series Is a Sequence in Disguise Take a series . Instead of confronting the whole infinite sum, build it up one term at a time. The n -th partial sum stops after the first n terms: the finite sum of the first n terms These partial sums are themselves a brand-new sequence , and each one is just the previous plus the next term: S_n = S_ n-1 + a_n . Watch one get built. 2. Convergence Is Partial-Sum Convergence Here is the definition that powers everything. A series converges exactly when its sequence of partial sums converges — and when it does, the limit is declared to be the value of the series. Below, the partial sums are plotted as a marching sequence. The convergent series climbs toward its limit and stops; the divergent harmonic series keeps climbing forever. 3. The n th-Term Test for Divergence One quick consequence saves a lot of work. If a series converges, consecutive partial sums must close in on the same limit, so their difference — which is exactly the term a_n — has to vanish: The logic: a_n = S_n - S_ n-1 . If then . So terms that don't die out are a guarantee the partial sums can never settle. Watch a series whose terms creep toward 1 instead of 0 . An infinite series is nothing more than the limit of its partial sums — series convergence is sequence convergence, wearing a different name.

This is the written version of the interactive lesson above. See the full Real Analysis course.