
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.
Help please !

This question seems trivial... Since addition is associative and commutative it shouldn't matter what order the terms are in...

I think the question is about infinite series. From Wikipedia: "Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any sum at all, including ∞ or −∞."

Let $\displaystyle a=\sum_{n=1}^\infty U_n$. Further, let M be such that for $\displaystyle S_1=\sum_{n=1}^M$ we have $\displaystyle \lvert S_1a\rvert<\varepsilon/2$. What can be said about $\displaystyle S_2=\sum_{n=1}^{M+N} U_n$? (Imagine that $\displaystyle M\gg N$.) All terms in $\displaystyle S_1$ are also in $\displaystyle S_2$. Also, N terms from the "tail" (i.e., remainder) of $\displaystyle S_1$ are in $\displaystyle S_2$, but the whole tail of $\displaystyle S_1$ is small, i.e., $\displaystyle <\varepsilon/2$. Therefore, $\displaystyle S_2$ is close to $\displaystyle a$.