Exam 3

1. A ball is dropped from a height of 100 m. On each bounce it attains a height four-fifth of the previous bounce. Find the total distance traveled by the ball (a) after after it has completed its fifth bounce; (b) after infinite bounces.
2. Find the sum:
   $$\sum_{n=0}^{\infty} \frac{2+3^{n}}{5^{n}}$$

3. Determine which of the following series converge; provide a brief explanation (and name the theorem you use if applicable).

   1. $$\sum_{n=1}^{\infty} \left[ \left(\frac{3}{4}\right)^{n} + \frac{1}{n} \right]$$

   2. $$\sum_{n=1}^{\infty} \frac{3}{(2n-1)^{2}}$$

   3. $$\sum_{n=1}^{\infty} \frac{1}{(2n)!}$$

   4. $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2n + 1}$$

4. Find the radius of convergence of the series
   $$\sum_{n=1}^{\infty} \frac{(2x)^{n}}{n}$$

5. Find the Taylor series of the function $f(x)=e^{x} \cos x$ about $x=0$ up to $x^{3}$ .
6. In Einstein's special relativity, the mass of an object moving with velocity $v$ is
   $$m=\frac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}$$
   where $m_{0}$ is a constant (the mass of the object when at rest) and $c$ is the speed of light.
   1. Find the Taylor series of $m$ about $v/c=0$ up to $(v/c)^{4}$ .
      Hint: let $(v/c)^{2}$ be $x$ .
   2. Find $v/c$ such that the ratio of the $(v/c)^{4}$ term to the $(v/c)^{2}$ term is $0.01$ .