convergent series

A convergent series is one whose endless addition genuinely lands somewhere: as you include more and more terms, the running total homes in on a definite finite number rather than drifting off or oscillating forever. That number is called the sum of the series.

In precise terms, the series sum a_n converges to S when its sequence of partial sums s_N = a_1 + ... + a_N satisfies lim s_N = S, with S finite. By the epsilon definition of a sequence limit, this means: for every epsilon > 0 there is an N_0 such that |s_N - S| < epsilon for all N >= N_0. Equivalently, by Cauchy's criterion, the series converges if and only if the tails get uniformly small — for every epsilon > 0 there is N_0 with |a_{m+1} + ... + a_n| < epsilon whenever n > m >= N_0.

Convergent series obey a clean algebra: if sum a_n = A and sum b_n = B converge, then sum (a_n + b_n) = A + B and sum (c a_n) = cA. But be careful — convergence of sum a_n forces a_n -> 0, yet the converse is false; the harmonic series has terms tending to zero and still diverges. Convergence is a statement about the partial sums, not merely about the size of individual terms.