Re: subseries

On 20 Jun 2006 20:14:57 -0700, MR <marcinrak@xxxxxxxxx> wrote:
Hey everyone

does a subseries by defintion have to preserve order of the original
series.

That is, if my original series sums the sequence {1,2,3,4,5,....}
can by definition a sub seires sum over the sequence {3,2,4,....} or
must it preserve the order?

The reason I ask, is the following definition of subseries in my calc
book:

Let f be a function whose domain is N and whose range is an infinite
subset of N, and assume that f is 1-to-1 on N. Let Sigma(a(n)) and
Sigma (b(n)) be two series s.t.

b(n)=a(f(n)) if n is in N

Then Sigma(b(n)) is said to be a subseries of Sigma(a(n))

As far as I can tell, this definition doesn't imply anything about the
order?

thanks
mr


I don't think I've heard the term "subseries" before, but as a general
principle order is very important for series: a conditionally convergent
series can be summed to _any_ value, including +/- infinity or no sum at all,
depending on order. So I would say that your book's definition, though not
totally absurd, conflates two important concepts: taking the sum of a sub
_sequence_ (where f is by general convention always taken to be an increasing
function) and rearranging a series (where f is any general bijection). In
fact, the book's definition can be reconstructed by first taking a
subsequence, and then rearranging (or indeed, the other way as well).

--
Ryan Reich
ryan.reich@xxxxxxxxx
sci.math
.

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