Series equivalence

[JellyFish]

I was wondering if anyone could make sense of this?

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

(*)? ai (where i goes from 0 to ?) converges to A if for every ? > 0 there is a K such that for all k ?K we have ? (? ai) – A ?< ? (where i goes from 0 to k).

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

(º) ? 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 < (? 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 (? ai) < U(where i goes from 0 to k).

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

 

[daon]

The second definition is merely saying the sum is bounded above by all numbers bigger than A, and bounded below by all number smaller than A.

If you take the superemum of all the L's and infimum of the U's, these quantities are equal and identically A.

The set of L's is (-infty, A), the set of U's is (A, infty). If the sum lies between these sets, it must be A.