We've got some pretty crazy functions in our real analysis class and we're trying to develop some mad plotting skills in our numerical analysis class. Why don't we try to mesh these together? So this question is in the new Analysis category, which is open to both my analysis classes. Here are a few fun functions to try:

$\displaystyle f(x) = \sum_{k=1}^{\infty} \sin(k^3x)/k^2$

$\displaystyle f(x) = \sum_{n=0}^{\infty} a^n \cos(b^nx)$

For this one, you might try several choices of $a$ and $b$. Be sure to consider the cases, $1< 1/a < b$ and $1 < b < 1/a$.Let $\{r_1,r_2,\ldots\}$ be an enumeration of the rational numbers. For each $r_n$, define

$$u_n = \begin{cases}

1/2^n & \text{ for } x>r_n \\

0 & \text{ for } x\leq r_n

\end{cases}$$

and then let

$$h(x) = \sum_{n=1}^{\infty} u_n(x).$$

Plot $h(x)$.

Any plotting tool of your choice is acceptable but plots produced by Python or Mathematica are particularly of interest.