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 ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{a}_n$ converges iffthe series ${\displaystyle \sum _{n=k+1}^{\mathrm{\infty }}}{a}_n$ converges.  Then the sums of both series are with

${\displaystyle \sum _{n=k+1}^{\mathrm{\infty }}}{a}_n={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{a}_n-{\displaystyle \sum _{n=1}^k}{a}_n.$(1)

Proof.  Denote the $k$th partial sum of ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ by $S_k$ and the $n$th partial sum of${\sum }_{n=k+1}^{\mathrm{\infty }}{a}_n$ by $S_n^{\prime }$.  Then we have

$S_n^{\prime }={\displaystyle \sum _{n=k+1}^{k+n}}{a}_n=S_{k+n}-S_k.$(2)

$1^{\circ }$.  If ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ converges, i.e.  ${lim}_{n\to \mathrm{\infty }}{S}_n:=S$  exists as a finite number, then (2) implies

$$\underset{n\to \mathrm{\infty }}{lim}{S}_n^{\prime }=\underset{n\to \mathrm{\infty }}{lim}{S}_{k+n}-\underset{n\to \mathrm{\infty }}{lim}{S}_k=S-S_k.$$

Thus ${\sum }_{n=k+1}^{\mathrm{\infty }}{a}_n$ converges and (1) is true.

$2^{\circ }$.  If we suppose ${\sum }_{n=k+1}^{\mathrm{\infty }}{a}_n$ to be convergent, i.e.  ${lim}_{n\to \mathrm{\infty }}{S}_n^{\prime }:=S^{\prime }$  exists as finite, then (2) implies that

$$\underset{n\to \mathrm{\infty }}{lim}{S}_n=\underset{n\to \mathrm{\infty }}{lim}{S}_{k+n}=\underset{n\to \mathrm{\infty }}{lim}(S_k+S_n^{\prime })=S_k+S^{\prime }.$$

This means that ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ is convergent and  $S=S_k+S^{\prime }$,  which is (1), is in .