# 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 $k$ be a natural number.  A series $\displaystyle\sum_{n=1}^{\infty}a_{n}$ converges iff the series $\displaystyle\sum_{n=k+1}^{\infty}\!a_{n}$ converges.  Then the sums of both series are with

 $\displaystyle\sum_{n=k+1}^{\infty}\!a_{n}\;=\;\sum_{n=1}^{\infty}a_{n}-\sum_{n% =1}^{k}a_{n}.$ (1)

Proof.  Denote the $k$th partial sum of $\sum_{n=1}^{\infty}a_{n}$ by $S_{k}$ and the $n$th partial sum of $\sum_{n=k+1}^{\infty}a_{n}$ by $S_{n}^{\prime}$.  Then we have

 $\displaystyle S_{n}^{\prime}\;=\;\sum_{n=k+1}^{k+n}\!a_{n}\;=\;S_{k+n}\!-\!S_{% k}.$ (2)

$1^{\circ}$.  If $\sum_{n=1}^{\infty}a_{n}$ converges, i.e.  $\lim_{n\to\infty}S_{n}:=S$  exists as a finite number, then (2) implies

 $\lim_{n\to\infty}S_{n}^{\prime}\;=\;\lim_{n\to\infty}S_{k+n}-\lim_{n\to\infty}% S_{k}\;=\;S\!-\!S_{k}.$

Thus $\sum_{n=k+1}^{\infty}a_{n}$ converges and (1) is true.

$2^{\circ}$.  If we suppose $\sum_{n=k+1}^{\infty}a_{n}$ to be convergent, i.e.  $\lim_{n\to\infty}S_{n}^{\prime}:=S^{\prime}$  exists as finite, then (2) implies that

 $\lim_{n\to\infty}S_{n}\;=\;\lim_{n\to\infty}S_{k+n}\;=\;\lim_{n\to\infty}(S_{k% }+S_{n}^{\prime})\;=\;S_{k}\!+\!S^{\prime}.$

This means that $\sum_{n=1}^{\infty}a_{n}$ is convergent and  $S=S_{k}\!+\!S^{\prime}$,  which is (1), is in .

Title finite changes in convergent series FiniteChangesInConvergentSeries 2013-03-22 19:03:10 2013-03-22 19:03:10 pahio (2872) pahio (2872) 9 pahio (2872) Theorem msc 40A05 SumOfSeriesDependsOnOrder RiemannSeriesTheorem RatioTestOfDAlembert