# limit of sequence as sum of series

If $U$ is the limit of a sequence

 $u_{1},\,u_{2},\,u_{3},\,\ldots$

of real or complex numbers, then $U$ can be expressed as the series sum

 $U=u_{1}+\sum_{i=1}^{\infty}(u_{i+1}-u_{i}).$

Proof. Let  $\displaystyle s_{n}:=u_{1}+\sum_{i=1}^{n-1}(u_{i+1}-u_{i})$.  We see that

 $s_{n}=u_{1}+\sum_{i=1}^{n-1}u_{i+1}-\sum_{i=1}^{n-1}u_{i}=u_{1}+\sum_{j=2}^{n}% u_{j}-\sum_{i=1}^{n-1}u_{i}=u_{n}$

for all  $n=1,\,2,\,3,\,\ldots$  Thus

 $u_{1}+\sum_{i=1}^{\infty}(u_{i+1}-u_{i})=\lim_{n\to\infty}s_{n}=\lim_{n\to% \infty}u_{n}=U,$

Q.E.D.

Title limit of sequence as sum of series LimitOfSequenceAsSumOfSeries 2013-03-22 17:28:21 2013-03-22 17:28:21 pahio (2872) pahio (2872) 4 pahio (2872) Theorem msc 40-00 SumOfSeries