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Summation Prove (Intro. to Algo. Open Course Episode 5) in Quick Sort

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When I prove the expectation of Randomized Quick Sort, I use the following fact. Therefore, I need to prove it.
Prove:

S n = \sum k = 2 n - 1 k l g k \leq 1 2 n 2 l g n - 1 8 n 2

Solve:

\sum k = 2 n - 1 k l g k < \int n - 1 2 k l g k d k < \int n 2 k l g k d k (k l g k > 0)

Integration by parts

S n < 1 2 k 2 l g k | n 2 - 1 2 \int n 2 k 2 1 k l n 2 d k = 1 2 n 2 l g n - 2 l g 2 - (l n 2 4 n 2 - l n 2) = 1 2 n 2 l g n - l n 2 4 n 2 + l n 2 - 2 < 1 2 n 2 l g n - l n 2 4 n 2 (l n 2 - 2 < 0) < 1 2 n 2 l g n - 1 8 n 2 (l n 2 > 1 2)

The problem is solved.

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阿里云maven镜像配置

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