Re: subseries


MR 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

Hi MR,

The definition in your book certainly makes no mention of order.
By the way, your book's definition is unnecessarily redundant. It
would suffice to say that f is 1 to 1 and then it would necessarily
follow that the range of f is infinite. Books are not always
consistent on terminology. Usually, subsequence means one of the
following:

(1) b(n) = a(f(n)) and f is strictly monotonic increasing (i.e.
m<n implies f(m) < f(n). Here order matters.

(2) b(n) = a(f(n)) and the map f from N to N is cofinal. This
means that for any natural number M there exists a natural number K
such that f(k) > or = M whenever k > or = K. Given the cofinality
condition here there is no particular gain in requiring that f be 1 to
1.

Your book's definition is inbetween. For most purposes it doesn't
matter which interpretation you ascribe to the word ``subsequence".
For example, the statement that if a sequence converges to a limit L
then any subsequence also converges to L is true under any of the three
interpretations.

Frankly your book's choice of notion for ``subsequence" seems
silly to me. If one wants maximum simplicity then go for deinition
(1). If one wants maximum generality then go for (2). The inbetween
condition chosen by your book is rather pointless. Out of curiosity,
which book is this?

Regards,
Mike

.

Relevant Pages

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