This post complements the previous TSP-related post by analyzing some asymptotic properties of the factorial.

*n*! vs *n*·2^{n}

Let’s start by defining both functions as products:

Extracting some factors:

We can see that

,

as it has the first product has the same number of factors and is greater or equal factor by factor. On the other hand, for *n* greater or equal than 9,

.

Combining this results, for *n* greater or equal than 9:

.

This is a conservative result, as the factorial is greater starting from *n* = 6.

#### A bound for the factorial values

One useful way of bounding sums of monotonically growing sequences whose terms follow a relatively simple function is by using the integral of the function. This applies the following identities:

(assuming and monotonically growing )

.

The factorial is the product of a simple sequence, not a sum, but we can use logarithms to bridge this gap:

.

So, applying the previously found bounds for sums, we get:

.

Integrating by parts the logarithm function:

.

Putting all together:

.

As the exponential function grows monotonically, we can apply it to all the members of the inequality:

.

Rearranging:

.

There are tighter bounds for the factorial, such as the Stirling’s series. But they are harder to prove 😀