Gabriel's Horn and Sum of Infinite Fractions: Contradiction or Connection?

[isukatphysics69]

I don't understand something, the sum n=1 until infinity of (1/n) is a divergent harmonic series meaning that its sum is infinite right?
After reading that i started thinking about the finite volume of the function (1/x) being revolved around the x-axis referred to as "Gabriels horn". They say that the area is getting so small as x -> infinity and that makes the volume finite after being revolved. Now they are saying that the sum of (1/n) from 1 to infinity is divergent, so they are taking these tiny fractions and summing them and saying that the sum will be infinite, that seems like it contradicts what they said about the finite volume. They are saying that an infinite amount of very small fractions will sum to infinity, but the very small area of 1/x as x-> infinity being revolved around the x-axis is going to produce a finite volume. Really confused here

 

[isukatphysics69]

That doesn't even make any sense , so youre telling me that an infinite of very small fractions being summed is going to be infinity but if i revolve an area about the x-axis from 1 to infinity i will get a finite volume? are you kidding me? this doesn't make any sense

 

[isukatphysics69]

So the inporper integral from 1 to infinity of 1/x is divergent but if i revolve that and create more area by doing so the volumes area is not infinite?

 

[haruspex]

When you take the volume of revolution, the area of each element (a disk thickness dx centred on the x axis) is proportional to y², not y. So now you have a sum like Σx^-2 instead of Σx^-1.
x^-2 gets smaller much faster than x^-1, and this makes all the difference.
In general Σx^-1-α converges for any α>0, no matter how slightly > 0, but not for α=0.

 

[isukatphysics69]

Ohh yes i am forgetting that the volume of rotation has to have pi*r^2 so you are taking the integral of the 1/x^2 not 1/x

 

[Mark44]

Gabriel's Horn offers an interesting paradox in that the volume is finite, but the surface area is infinite.The implication here is that if you could fill it up with a finite amount of paint that presumably would cover the inside completely, yet it would take an infinite amount of paint to cover the outside of the horn.

 

[Mark44]

the sum n=1 until infinity of (1/n)

BTW, if you're going to ask questions about infinite series, you should probably take the time to learn how to write them. Investing about 10 minutes will take you a long way.

We have a tutorial here -- https://www.physicsforums.com/help/latexhelp/

Here's the series you're asking about:
$$\sum_{n = 1}^\infty \frac 1 n$$

Here's my LaTeX script, unrendered, for the above:
$$\sum_{n = 1}^\infty \frac 1 n$$