Third Exam

1. (10 points) A ball is dropped from a height of 50 m. On each bounce it attains a height $\frac{4}{5}$ 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: (10 points)
   $$\sum_{n=0}^{\infty} \frac{2+3^{n}}{5^{n}}$$

3. (32 points) Determine which of the following series converge; provide a brief explanation (and name the theorem you use if applicable).

   1. $$\sum_{n=1}^{\infty} \frac{\ln n}{n}$$

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

   3. $$\sum_{n=1}^{\infty} \left[ \left(\frac{3}{4}\right)^{n} + \frac{1}{n^{2}} \right]$$

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

4. (10 points) Find the radius of convergence of the series
   $$\sum_{n=1}^{\infty} \frac{ (2n)! x^{n}}{(n!)^{2}}$$

5. (10 points) Find the Taylor series of the function $f(x)=\sqrt{1-2x}$ about $x=0$ up to $x^{3}$ .
6. (10 points) Find the Taylor series of the function $f(x)=e^{x} \cos x$ about $x=0$ up to $x^{3}$ .
7. (10 points) Find the Taylor series of the function $f(x) = 1/x$ about $x=1$ up to $(x-1)^{4}$ .

8. (15 points) Estimate the value of $\int_{0}^{1} (\sin x/x) dx$ by approximating the function $\sin x/x$ with a Taylor polynomial of degree $4$ .