1.2. Useful Results#
In this section, we’ll cover several useful results that will come up repeatedly throughout this book. You should remember these results from calculus courses (even if you can’t quite recall the proofs).
1.2.1. Bounding with the Exponential#
The first result is an extremely trivial result about how to use the exponential to bound:
Lemma 1.1 (Bounding with exponential)
Let
Proof. Let
Note that
Note that the second derivative is positive for any
Further, note that
Then:
1.2.2. Integration by Parts#
To define the integration by parts formula, we need to introduce a concept known as differentiability classes:
Definition 1.12 (Differentiability class
Let
Next is the popular integrations by parts formula:
Lemma 1.2 (Integration by parts)
Suppose that
Proof. Note that by the Chain Rule:
Integrating both sides with respect to
After rearrangement.
Taking the difference of the right hand side for
1.2.3. Stirling’s approximation#
Finally, we get to the popular Stirling’s approximation formula:
Lemma 1.3
Suppose that