• closed class words - words with fixed memberships (prepositions, conjunctivas, etc.); not being created or added much, used for grammatical function
• open class words - words that are set as content, and are focused on content

For some \(a \in \mathbb{F}\), we define \(a^m\) to be \(a\) multiplied with itself \(m\) times.

additional information

• \((a^m)^n = a^{mn}\)
• \((ab)^m = a^mb^m\)

a power series centered at \(a\) is defined with \(c_{n} \in \mathbb{R}\), whereby:

\begin{equation} f(x) = \sum_{n=0}^{\infty} c_{n}(x-a)^{n} \end{equation}

meaning it is written as \(c_0 + c_1(x-a) + c_2(x-a)^{2} + c_3 (x-a)^{3} + \cdots\)

radius of convergence

• there is a radius of convergence \(R \geq 0\) for any power series, possibly infinite, by which the series is absolutely convergent where \(|x-a| < R\), and it does not converge when \(|x-a| > R\) , the case where \(|x-a| = R\) is uncertain
• ratio test: if all coefficients \(c_{n}\) are nonzero, and some \(\lim_{n \to \infty} \left| \frac{c_{n}}{c_{n+1}} \right|\) evaluates to some \(c\) — if \(c\) is positive or \(+\infty\), then that limit is equivalent to the radius of convergence
• Taylor’s Formula: a power series \(f(x)\) can be differentiated, integrated on the bounds of \((a-R, a+R)\), the derivatives and integrals will have radius of convergence \(R\) and \(c_{n} = \frac{f^{(n)}(a)}{n!}\) to construct the series

linear combinations of power series

When \(\sum_{n=0}^{\infty} a_{n}\) and \(\sum_{n=0}^{\infty} b_{n}\) are both convergent, linear combinations of them can be described in the usual fashion:

We can now use power series to also solve differential equations.

\begin{equation} \dv{x}{t} = 0; x(0)=1 \end{equation}

We wish to have a power-series solution of shape:

\begin{equation} x(t) = \sum_{k=0}^{\infty }a_{k}t^{k} \end{equation}

We want to find the coefficients \(a_{k}\). If you can find such a function that fits this form, they both 1) converge and 20 behave the same way as \(e^{x}\) does in Simple Differential Equations.

analytic functions

Functions which can be described with a power series are called analytic functions.

the power set of \(A\) is the set of all subsets of \(A\). that is, the power set of \(\qty {0,1}\) is \(\qty {\emptyset, \qty {0}, \qty {1}, \qty {0,1}}\).