finite changes in convergent series
The following theorem means that at the beginning of a convergent series, one can remove or attach a finite amount of terms without influencing on the convergence of the series – the convergence is determined alone by the infinitely long “tail” of the series. Consequently, one can also freely change the of a finite amount of terms.
Theorem. Let be a natural number. A series converges iff the series converges. Then the sums of both series are with
(1) |
Proof. Denote the th partial sum of by and the th partial sum of by . Then we have
(2) |
. If converges, i.e. exists as a finite number, then (2) implies
Thus converges and (1) is true.
. If we suppose to be convergent, i.e. exists as finite, then (2) implies that
This means that is convergent and , which is (1), is in .
Title | finite changes in convergent series |
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Canonical name | FiniteChangesInConvergentSeries |
Date of creation | 2013-03-22 19:03:10 |
Last modified on | 2013-03-22 19:03:10 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 9 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 40A05 |
Related topic | SumOfSeriesDependsOnOrder |
Related topic | RiemannSeriesTheorem |
Related topic | RatioTestOfDAlembert |