Analyzing Randomly Descending Sequences

Finals week just ended for me so I'm hoping to post some new cool math problems in the next few days.  In this blog post, I present a math problem that is very easy to understand what it is asking, but not so easy to solve.  Regardless, I will present my solution, which yielded a very interesting result.
You construct a sequence \( k_1, k_2, ..., k_m \) by randomly picking an integer from the set \( 0 \leq k_1 < n \).  After picking \( k_1 \), you repeat the process by picking \( k_2 \) randomly from \( 0 \leq k_2 < k_1 \) and repeat the process until you can no longer pick any numbers (when \( k_m = 0 \)).  Find a function for \( f(n) \), where \( f(n) \) is the expected value of the length of this sequence.


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