The Stirling Formula

Wallis's Formula

Let's start with a definite integral:

We can derive the following result:

Using integration by parts, where , we get:

Thus:

We get:

Therefore:

For even , i.e., :

For odd , i.e., :

Here, we need the double factorial:

Thus:

Since for , we have:

Substituting the above formulas:

We can examine the difference between the bounds for :

Note: Here

For the inequality:

Multiply all terms by :

We can further examine the difference between the bounds for :

Thus, we obtain Wallis's Formula:

This formula introduces Wallis's result in preparation for deriving Stirling's formula.

Stirling's Formula

Stirling's formula reveals a relationship between , , and .

As approaches infinity, is approximately .

First, introduce a limit:

Proof:

which is equivalent to proving:

We have:

QED.

Thus,

We would like to verify:

Let’s test this.

Using the ratio test:

Although the limit of the expression is , it is always approaching from below, so it is still greater than 1, indicating that is divergent.

Mathematicians have attempted various approaches and found:

To prove that converges, use the ratio test again:

Compare this with:

and .

Thus, compare:

with 1.

Specifically, compare with :

Since:

Note: The right side follows from the inequality (for ).

The left side follows from: , where is considered as a whole. This is high school knowledge.

Because is a concave function:

Therefore:

Thus, is monotonically decreasing and , so it must converge, i.e., it has a limit.

Let:

Then, we need to find this limit.

From the earlier Wallis formula:

Thus, Wallis' formula becomes:

Using:

Substitute into:

Solve for:

Thus, as approaches infinity:

or equivalently:

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