6.5 Chapter Summary

This chapter focused on methods for approximating functions.

Interactive Tools:¶

1. Taylor Series Visualizer - Use this tool to visualize Taylor series.

Taylor Series¶

Definitions and examples are all available in Section 6.1.

1. Given a smooth function $f$, its Taylor series expansion about $x_*$ is:

   $$f(x) = \sum_{n = 0}^{\infty} \frac{1}{n!} f^{(n)}(x_*) (x - x_*)^n$$

   (1)

   where $f^{(n)}(x) = \frac{d^n}{dx^n} f(x).$

2. The $m^{th}$ order polynomial approximation to $f$ about $x_*$ is the polynomial:

   $$f(x) \simeq \tilde{f}_m(x) = \sum_{n = 0}^{m} \frac{1}{n!} f^{(n)}(x_*) (x - x_*)^n$$

   (2)

   • The error in the $m^{th}$ order approximation is $\mathcal{O}((x - x_*)^{m+1})$

   • The linear and quadratic approximations are:

   $$\begin{aligned} & f(x) \simeq f(x_*) + f'(x_*) (x - x_*) \\ & f(x) \simeq f(x_*) + f'(x_*) (x - x_*) + \frac{1}{2} f''(x_*) (x - x_*)^2. \end{aligned}$$

   (3)

3. The Taylor expansion of the exponential is:

   $$e^x = \sum_{n = 0}^{\infty} \frac{1}{n!} x^n.$$

   (4)

4. The linear approximation to the logarithm near 1 is:

   $$\log(1 +x) \simeq x.$$

   (5)

Exponential Approximation:¶

Definitions and examples are all available in Section 6.2.

1. The limit definition of the exponential is:

   $$\lim_{n \rightarrow \infty} \left(1 + \frac{1}{n} x \right)^n = e^x.$$

   (6)

2. If incidents occur randomly in time, with time intervals $T$ between successive incidents such that:

   • $T$ is continuously distributed,

   • disjoint time intervals are independent, and

   • the chance an incident occurs in a time interval $[t,t']$ is only a function of the duration of the interval, and is $\mathcal{O}(t' - t)$ when $t' - t$ is small,

   then $T$ is exponentially distributed.

Factorial Approximation:¶

Definitions and examples are all available in Section 6.3.

1. Stirling’s Approximation:

   $$n! \approx \sqrt{2 \pi e} \left(\frac{n}{e} \right)^{n + \frac{1}{2}}.$$

   (7)

Limiting Distributions:¶

Definitions and examples are all available in Section 6.4.

1. A random variable $X$ is Poisson distributed with parameter $\lambda$ if:

   $$X \in \{0,1,2,3,...,\infty\}, \quad \text{PMF}(x) = e^{-\lambda} \frac{\lambda^x}{x!}.$$

   (8)

   • The Law of Small Numbers: If $X_n \sim \text{Binomial}(n,p_n)$ and $\lim_{n \rightarrow \infty} \mathbb{E}[X_n] = \lim_{n \rightarrow \infty} n p_n = \lambda$, then $X = \lim_{n \rightarrow \infty} X_n$ is Poisson distributed with parameter $\lambda$.

   • If incidents occur randomly in time, where the times between consecutive incidents are independent, identically, and exponentially distributed, then the total number of incidents in a fixed time interval is Poisson distributed with parameter proportional to the length of the interval.

2. A random variable $Z$ is standard normal if:

   $$Z \in (-\infty,\infty), \quad \text{PDF}(z) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} z^2}$$

   (9)

   • If $X_n \sim \text{Binomial}(n,p)$, and $Z_n$ equals $X_n$ after standardizing, then $Z_n$ converges to $Z$ as $n$ diverges.

   • Binomial/choose coefficients are, for large $n$, bell shaped as a function of their second input, and are approximately proportional to a normal curve.