Introduction

$\ell n N!$ is the starting point for this derivation, rather than N factorial (N!) itself.

$$\ell n N! = \ell n N +\ell n (N-1) + \ell n (N-2) \cdots$$

which can be written as

$$\ell n N! = \sum_{j=0}^{j=N-1}\ell n (N-j)$$

We now write this backwards:

$$\ell n N! = \sum_{j=1}^{j=N}\ell n (j)$$

which covers the same territory.

 

Figure 1: The logarithmic factorial sum, shown explicitly.

Next, we convert this sum to an area by constucting horizontal bridges (as shown) where the width of each rectangle is going to turn out to be one (1)! This means that the height and the area are synonymous!

 

Figure 2: The logarithmic factorial sum, reversed.

We then have

$$\ell n N! = \sum_{j=1}^{j=N}\ell n (j) [(j+1)-j]$$

as an area, and we rewrite this as

$$\ell n N! = \sum_{j=1}^{j=N}\ell n (j) \Delta j$$

preparatory to making the histogram to continuous functional area transformation taught in first year calculus.

 

Figure 3: The logarithmic factorial sum, converted to a histogram.

We then have

$$\ell n N! = \int_{j=1}^{j=N}\ell n (j) \delta j$$

which is trivially integrable to give

$$\ell n N! = \left .j\ell n j-j\right \vert _{j=1}^{j=N}$$

which evaluates to

$$\ell n N! = N\ell n N-N- ( 1\ell n 1-1)$$

which is, in the limit N much larger than 1

$$\ell n N! = N\ell n N-N$$