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.
Proof. Denote the th partial sum of by and the th partial sum of by . Then we have
. 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|
|Date of creation||2013-03-22 19:03:10|
|Last modified on||2013-03-22 19:03:10|
|Last modified by||pahio (2872)|