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

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

Sn=∑k=2n−1klgk≤12n2lgn−18n2

S_{n}=\sum_{k=2}^{n-1}klgk\le\frac{1}{2}n^2lgn-\frac{1}{8}n^2
Solve:

∑k=2n−1klgk<∫n−12klgkdk<∫n2klgkdk(klgk>0)

\sum_{k=2}^{n-1}klgk0)
Integration by parts

Sn<12k2lgk|n2−12∫n2k21kln2dk=12n2lgn−2lg2−(ln24n2−ln2)=12n2lgn−ln24n2+ln2−2<12n2lgn−ln24n2(ln2−2<0)<12n2lgn−18n2(ln2>12)

S_{n}\frac{1}{2})

The problem is solved.

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

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