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 \(x \in \mathbb R\). Then:

\[ 1 + x \leq \exp(x)\]

Proof. Let \(f(x) = \exp(x) - 1 - x\). Notice that the derivative \(f'(x) = \exp(x) - 1\), and the second derivative is \(f''(x) = \exp(x)\).

Note that \(\exp(x) - 1 = 0\) when \(x = 0\), so \(f(x)\) has a critical point at \(x^* = 0\).

Note that the second derivative is positive for any \(x\), so \(x^* = 0\) is a global minimum of \(f(x)\).

Further, note that \(f(x) \geq 0\) for all \(x\).

Then:

\[\begin{split} 0 \leq f(x) &= \exp(x) - 1 - x \\ \Rightarrow 1 + x &\leq \exp(x).\end{split}\]

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 \(C^m\))

Let \(\mathcal X \subset \mathbb R\) be an open set, and let \(f : \mathcal X \rightarrow \mathbb R\) be a function. The function \(f\) is said to be of differentiability class \(C^m\) on \(\mathcal X\) for \(m \in \mathbb N\) if the derivatives \(f^{(k)}\) exist and are continuous on \(\mathcal X\) for all \(k \in [m]\).

Next is the popular integrations by parts formula:

Lemma 1.2 (Integration by parts)

Suppose that \(u, v \in C^1 : \mathbb R \rightarrow \mathbb R\) are continuously differentiable functions. Then:

\[\begin{split} \int u(x)\frac{\text d}{\text d x}[v(x)] \text d x &= u(x) v(x) - \int v(x) \frac{\text d}{\text x}[u(x)] \text d x \\ \int_a^b u(x)\frac{\text d}{\text d x}[v(x)] \text d x &= u(b) v(b) - u(a)v(a) - \int_a^v v(x) \frac{\text d}{\text x}[u(x)] \text d x\end{split}\]

Proof. Note that by the Chain Rule:

\[ \frac{\text d}{\text d x} [u(x) v(x)] = v(x) \frac{\text d}{\text d x}[u(x)] + u(x)\frac{\text d}{\text d x}[v(x)].\]

Integrating both sides with respect to \(x\):

\[\begin{split} \int \frac{\text d}{\text d x} [u(x) v(x)]\text d x = u(x)v(x) &= \int v(x) \frac{\text d}{\text d x}[u(x)]\text d x + \int u(x)\frac{\text d}{\text d x}[v(x)] \text d x \\ \Rightarrow \int u(x)\frac{\text d}{\text d x}[v(x)] \text d x &= u(x)v(x) - \int v(x) \frac{\text d}{\text d x}[u(x)]\text d x\end{split}\]

After rearrangement.

Taking the difference of the right hand side for \(\big |_{x = a}^b\) gives the second result of the desired lemma statement.

1.2.3. Stirling’s approximation

Finally, we get to the popular Stirling’s approximation formula:

Lemma 1.3

Suppose that \(n \in \mathbb n\). Then:

\[ n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n\]