Re: subseries, correction to previous


Mike wrote:
I just realized that I when I pontificated about the way to define
subsequence, I had forgotten that the point was to get to subseries. I
retract my criticism of your book. For the purpose of defining
subseries one does want the 1 to 1 condition for f (order is not
needed). For issues involving simple convergence of sequences,
however, the cofinal condition is all one needs.

Mike


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

alright, so you say tha the subseries need not preserve the order but:
1) a subsequence does - correct?
2)I also have a definition for a rearrangement of a series, and it is:
"Let f be a function whose domain is N and whose range is 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))
then Sigma(b(n)) is a rearrangement of Sigma(a(n))

then what is the difference b/w a rearrangement of a series and a
subseries?

mr

.

Relevant Pages

  • Re: subseries
    ... does a subseries by defintion have to preserve order of the original ... is the following definition of subseries in my calc ... subsequence means one of the ... the statement that if a sequence converges to a limit L ...
    (sci.math)
  • Re: subseries, correction to previous
    ... subsequence, I had forgotten that the point was to get to subseries. ... retract my criticism of your book. ... the cofinal condition is all one needs. ...
    (sci.math)
Quantcast