slower convergent series
Theorem.
If
(1) |
is a converging series with positive , then one can always form another converging series
such that
(2) |
Proof. Let be the sum of (1), the partial sum of (1) and the corresponding remainder term. Then we have
We set
Then the series fulfils the requirements in the theorem. Its are positive. Further, it converges because its partial sum is equal to which tends to the limit as since ; this implies also (2).
Title | slower convergent series |
---|---|
Canonical name | SlowerConvergentSeries |
Date of creation | 2013-03-22 15:08:24 |
Last modified on | 2013-03-22 15:08:24 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 11 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 40A05 |
Related topic | SlowerDivergentSeries |
Related topic | NonExistenceOfUniversalSeriesConvergenceCriterion |