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pseries
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 pseries. The volume > of revolution formed by revolving y=1/x; x>1 about the xaxis is finite but has > an infinite surface area. So, theoretically, you could fill it with paint but > you couldn't paint it? > > JUI don't see the relationship with pseries.
The relationship with pseries 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=2series in there someplace. Similarly the surface area is a pseries with p=1, and so divergent.
> JUOn 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: JUI 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.
GHYou 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 pseries is a nice example.
Geoff H



