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Cauchy criterion for convergence
A series in a Banach space is convergent iff for every there is a number such that
holds for all and .
Proof:
First define
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 convergent iff
Type of Math Object:
Theorem
Major Section:
Reference
Mathematics Subject Classification
40A05 Convergence and divergence of series and sequences- Forums
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new problem: Curve fitting using the Exchange Algorithm. by jeremyboden
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Corrections
the the by yark ✓
use \cdots by Mathprof ✓
contains own proof by rspuzio ✓
Not quite by rm50 ✓
Contains own proof by rm50 ✓
use \cdots by Mathprof ✓
contains own proof by rspuzio ✓
Not quite by rm50 ✓
Contains own proof by rm50 ✓
Versions
(v14) by mathwizard 2013-03-22



Comments
Unproven
The entry appears as unproven although it is not.