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Topic: p-series
Replies: 0

 Geoff Hagopian Posts: 97 Registered: 12/6/04
p-series
Posted: Mar 12, 1998 7:01 PM

Jerry Uhl wrote:

> 3/9/98, GH wrote:
> >There's another interesting paradox which is related to the p-series.
The volume
> of revolution formed by revolving y=1/x; x>1 about the x-axis is
finite but has
> an infinite surface area. So, theoretically, you could fill it with
paint but
> you couldn't paint it?
>
> JU----I don't see the relationship with p-series.

The relationship with p-series I see is this: The volume is
pi*int(1/x^2,x,1,infinity) =
pi*lim(lim(sum(1/(1+kN/n)^2,k,1,n),n,infinity),N,infinity) and I think
there's a
p=2-series in there someplace. Similarly the surface area is a p-series
with p=1,
and so divergent.

> JU----On the otherhand, if I can fill this all the way to the tip,
then the I
> must be able to make the paint arbitrarily thin. If I am be able to
make the
> paint arbitrarily thin, then I can paint it with any positive amount
of paint. I
> use have the paint to paint 1<=x<=2, then half the remaining paint to
paint 2
> < = x <= 3, etc.
> ----------------------------------------------------------------------

Does this resolve the paradox? It seems that "if I can fill this all
the way to
the tip" is a big "if..."

In another communication Jerry writes:
JU----I agree that the Riemann zeta function is important in mathematics
and
science. But none of the great applications Mark gives here seem
important in
freshman calculus.

GH----You seem to be defining your calculus curriculum on a "need to
know"
basis; that is, you will only teach topics it has been demonstrated that
students need to know - which is, it seems to me, a deflationary in that
it ultimately reduces the calculus curriculum down to it's kernel: the
definition of the limit.
I take an approach which ads a variable list of topics to the basic
canon: these
additional topics might inlcude the Riemann Zeta function one semester
or, as I am
doing this semester, Fibonacci series.
Also, if the kernel of calculus is the concept of the limit, then it
seems to me it benefits students to experience the infinite and
infinitessimal in various situations of which the p-series is a nice
example.

-Geoff H