/sci/ - Science & Math

Say that I have a non-increasing sequence [math](a_n)[/math] which converges absolutely to 0, and in fact I know that [math]\sum_{n=1}^\infty |a_n| = 1[/math]. If I only take the first [math]N[/math] terms of this sequence, how close to it (in terms of, say, the norm on [math]\ell^1[/math]) am I guaranteed to be? That is, what can I say about the tail term [math]\| (a_n) - (a_1, \ldots, a_N, 0, \ldots ) \|_1
= \sum_{n=N+1}^\infty |a_n|[/math], and how can I characterize it?