Originally Posted by

**CoraGB** This problem has left me scratching my head:

The modern (and Archimedean!) meaning of “the series Σ ai (where i goes from 0 to ∞)converges to A" isusually captured by a definition like:

(*)Σ ai (where i goes from 0 to ∞)converges to A if for every ε > 0 there is a K such that for all k ≥K wehave │ (Σ ai) – A │< ε (where i goes from 0 to k).

Archimedes himself would probably have said something more along the following lines:

(º) Σ ai (where i goes from 0 to ∞)converges to A if both

(1) for every L < A there is a K such that for all k ≥K we have L < (Σ ai) (where i goes from 0 to k),

and

(2) for every U > A there is a K′such that for all k ≥ K′we have (Σ ai) < U(where i goes from 0 to k).

Explain, in detail, why these two definitions are actually equivalent.