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 order 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 connected with

(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 $S_n'$ . Then we have

(2)

$2^\circ$ . If we suppose $\sum_{n=k+1}^\infty a_n$ to be convergent, i.e. $\lim_{n\to\infty}S_n' := S'$ exists as finite, then (2) implies that $$\lim_{n\to\infty}S_n \;=\; \lim_{n\to\infty}S_{k+n} \;=\; \lim_{n\to\infty}(S_k+S_n') \;=\; S_k\!+\!S'.$$ This means that $\sum_{n=1}^\infty a_n$ is convergent and $S = S_k\!+\!S'$ , which is (1), is in force.