Written in haskell (code here) and compiled with Haste-compiler (v.0.5.0) (updated!).

Rectangular : a_n = 1/n if n is even else 0
Triangle : 1, 0, -1/9, 0, 1/25,...
sawTooth : 1/n
leafy : Im( iⁿ) / ((n+1) log (n+1))
wavy : 1/ n log(n)
play0: (-1)ⁿ / log (n)
play1 :: 1 / n (when n = 7k+1)