# Math Help - Rearrangement of Terms

1. ## Rearrangement of Terms

Let sigma Un be a convergent series, and let sigma Vn be a rearrangement of it. In the rearrangement, suppose that no term of the original series is moved more than N places from its original position, where N is a fixed number. Show that the new series is convergent and has the same value as the old one.

4. Let $a=\sum_{n=1}^\infty U_n$. Further, let M be such that for $S_1=\sum_{n=1}^M$ we have $\lvert S_1-a\rvert<\varepsilon/2$. What can be said about $S_2=\sum_{n=1}^{M+N} U_n$? (Imagine that $M\gg N$.) All terms in $S_1$ are also in $S_2$. Also, N terms from the "tail" (i.e., remainder) of $S_1$ are in $S_2$, but the whole tail of $S_1$ is small, i.e., $<\varepsilon/2$. Therefore, $S_2$ is close to $a$.