The Expected Number of Cycles in a Permutation

I recently came across a really interesting result that counts the expected number of cycles in a permutation of size n.
It is really cool because I often looked at cycles in permutations, and wondered how many one could get as n grows large. The answer may be surprising but, should make intuitive sense.
There are 2 approaches shared below - one is based on generating functions (and loosely symmetric functions), and the other is quite elementary.

The content below contains my adaptation of a proof I read and also a different approach which is my own.

Find the Proof Here

I hope you enjoyed reading this!!!

Credit: Stanley Richard. Enumerative Combinatorics. Cambridge University Press, 2012.