opticaltempest
New member
- Joined
- Nov 19, 2005
- Messages
- 48
Do my solutions look correct?
My solution to (a)
The height of the stacked spheres can be represented by the infinite series
\(\displaystyle \L
2\sum\limits_{n = 1}^\infty {\frac{1}{{n^{\frac{1}{2}} }}}\)
The above infinite series is a divergent p-series which implies that the stack
of the spheres is infinitely tall.
(b)
The surface area of a sphere can be found by using the formula \(\displaystyle \L
4\pi r^2\).
The surface area of our infinitely tall stack of spheres is
\(\displaystyle \L
4\pi \sum\limits_{n = 1}^\infty {\left( {\frac{1}{{n^{\frac{1}{2}} }}} \right)} ^2 = 4\pi \sum\limits_{n = 1}^\infty {\frac{1}{n}}\)
The above infinite series is a divergent p-series which implies that the spheres
have an infinite amount of surface area.
(c)
The volume of the spheres is given by the infinite series below
\(\displaystyle \L
\frac{4}{3}\pi \sum\limits_{n = 1}^\infty {\left( {\frac{1}{{n^{\frac{1}{2}} }}} \right)} ^3 = \frac{4}{3}\pi \sum\limits_{n = 1}^\infty {\frac{1}{{n^{\frac{3}{2}} }}}\)
The above series is a convergent p-series which implies that the weight of our
infinitely tall stack of spheres is a finite amount.
