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 !
February 11th 2011, 04:19 AM
This question seems trivial... Since addition is associative and commutative it shouldn't matter what order the terms are in...
February 11th 2011, 04:30 AM
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 −∞."
February 11th 2011, 06:23 AM
Let . Further, let M be such that for we have . What can be said about ? (Imagine that .) All terms in are also in . Also, N terms from the "tail" (i.e., remainder) of are in , but the whole tail of is small, i.e., . Therefore, is close to .