Cauchy criterion for convergence
A series ∑∞i=0ai in a Banach space (V,∥⋅∥) is http://planetmath.org/node/2311convergent iff for every ε>0 there is a number N∈ℕ such that
∥an+1+an+2+⋯+an+p∥<ε |
holds for all n>N and p≥1.
Proof:
First define
sn:= |
Now, since is complete, converges
if and only if it is a Cauchy sequence
, so if for every there is a number , such that for all holds:
We can assume and thus set . The series is iff
Title | Cauchy criterion for convergence |
---|---|
Canonical name | CauchyCriterionForConvergence |
Date of creation | 2013-03-22 13:22:03 |
Last modified on | 2013-03-22 13:22:03 |
Owner | mathwizard (128) |
Last modified by | mathwizard (128) |
Numerical id | 14 |
Author | mathwizard (128) |
Entry type | Theorem |
Classification | msc 40A05 |