As you go to higher levels, that "infinite surface area" thing starts to become difficult to manage. ;-)
Sponge has infinite area and 0 volume.
Sierpinski's triangle has infinite perimiter and 0 area.
I wonder why it takes three dimensions before people start getting upset.
On the other hand, if you take 1/x^2 as the integrand, you get the desired effect: its antiderivative is -1/x, so the integral of 1/x^2 from x=1 to infty is finite (actually, it's 1).
By the way: For good fun with convergent series, take a look at the problem of escaping a lion in a circular arena at equal maximum speeds. Here's an example link to both question and answer: http://puzzling.stackexchange.com/questions/8140/escaping-a-...
IIRC, there is an even more fascinating story associated to the history of this problem, as the first published solution was actually incorrect.
Read the paradox description. To fill the horn with paint you need a finite amount. To paint the outside of the horn you need an infinite amount.
That is the paradox. The surface area of the inside is equal to the outside, yet one side requires a finite amount of paint while the other side requires an infinite amount.
At least that's how I understand the paradox.
The intended "paradox" is that the surface area is infinite but the volume is finite.
>Since the Horn has finite volume but infinite surface area, it seems that it could be filled with a finite quantity of paint, and yet that paint would not be sufficient to coat its INNER surface
An anti-derivative of 1/x is ln |x| (very different from |ln x|) when the domain excludes x=0. By the fundamental theorem of calculus, you can use it to integrate over intervals that exclude x=0. (On positive-real intervals F(x)=ln(x) works, and on negative-real intervals F(x)=ln(-x) works. In either case, F(x)=ln|x| is an equivalent formula. If you try a domain that includes x=0, you risk not just a problem with technicalities but also the practical problem that F(x)=ln|x| and G(x)=ln|x|+sgn(x) look no different...)
├┘│3│ │ │
├─┘-│ │ │
│ 64│ │ │
├───┘ │ │
│ │ │
│ 3/16 │ │
│ │ │
│ 3/4 │
edit: better text art
10/3(3.333...) is 3 + 3/10 + 3/100/ + 3/1000 + 3/10000...
π(3.1415...) is 3 + 1/10 + 4/100 + 1/1000 + 5/10000...
Its quite easy to see that both of these series will be finite, and many numbers, lets say 4 or 3 + 5/10 will be greater than either of them.
9 + 0.9 + 0.09 + 0.09 + 0.009...
 Electromagnetism, classical mechanics, etc
Construct a set of discs in R^2 s.t. no infinite straight line can be drawn without intersecting at least one disc, and the sum of the areas of all the discs is finite.
(Awarded 1 point of 10 for being unable to specify the precise piecewise-linear spiral function)