2004 #01 and 41

Start of a new series. A nice fraction and a holy craptastic function thing. Wow!

Difficulty: Simple.

Difficulty: Relies on a bunch of work and a couple AHA! moments.  Tough.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2004

answer 01, answer 41.