how to prove this express is larger than the other one?

Question

Can anyone give me a hint on prove this
Summation from k=1 to n
log(n!/(n-k)!)*log(n-k)

is larger than this

Summation from k=2 to n
log((k-1)!)*log(k+1)

Answer 1

The first one is not necessarily larger than the second.
First of all, you have a problem when k = n, since it involves finding log(n-k).

Secondly, for n = 5 (arbitrary)
the first expression sums to 8.36 (omitting k = 5)
the second sums to 9.54

Thirdly, the second summation is not dependent on the starting value of k, since log((k-1)!) = 0 when k=1.

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