1. ## infinite series

Imagine you are stacking an infinite number of spheres of decreasing radii on top of each other. The radii of the spheres are 1m, 1/2^(1/2)m, 1/3^(1/2)m, etc. . . The spheres are made of a material that weighs 1 newton per cubic meter. . .

1.) How high is the infinite stack of spheres

2.) What is the total surface area of all the spheres in the stack

3.) Show that the weight of the stack is finite.

2. 1)

The following series:

$\displaystyle 2 \sum_{1 }^{ \infty } \left ( { \frac{ 1}{n } } \right ) ^{1/2}$

is divergent. The stack would be infinitely high.

2)

Surface Area = 4*pi*r^2

-> Total surface area = $\displaystyle 4 \pi \sum_{1 }^{ \infty } \left ( \left ( \frac{1 }{n } \right ) ^{1/2} \right ) ^{2} = 4 \pi \sum_{1 }^{ \infty } \frac{1 }{ n}$

This is the famous harmonic series. Also divergent. So surface area is infinite.

3) You can probably see where this one will lead. Since volume is proportional to r^3, you will have an overharmonic series which will converge:

$\displaystyle \frac{4 }{3 } \pi \sum_{1 }^{ \infty } \left ( \left ( \frac{1 }{n } \right ) ^{1/2} \right ) ^{3} = \frac{ 4}{ 3} \pi \sum_{1 }^{ \infty } ( \frac{1 }{ n} )^{ \frac{ 3}{2 } } = \frac{ 4}{ 3} \pi \zeta (3/2) \approx 10.94$ cubic meters

-> Weight = 10.94 newtons

Thus, infinite height, infinite surface area, finite weight.

3. thank you. I had a feeling that was going to be how it worked out, but i wasn't positive on how to actually show it. Thanks

sorry one more question. I understand the p-series test has (1/n)^(3/2) converging, but i dont understand to what value it converges or how you came up with 10.94 meters

4. Okay so check out the Riemann Zeta function as given by wikipedia:

So in your case s = 3/2. Wikipedia gives Zeta(3/2) = 2.612

So (4/3)*Pi*2.612 = approx 10.94

5. o ok thank you