Partial sum of n! (books or PDFs)

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Hello, I am looking for materials (books or PDFs) on how to derive the partial sum of n! Unlike other partial sums this does not seem so straightforward to me: https://mathworld.wolfram.com/FactorialSums.html I am trying to find material on how (2) equals (3) (or 4) in this case. I have already gone trough the reference material and could not find the method of derivation.