User:Eben Stone/sandbox

TODO: Learning objectives as statements

TODO: Intro/explanation
 * Look-Ahead
 * Series
 * Discrete Mathematics

Introduction


Think of a sequence as many numbers that are arranged in a particular order. Consider the following example of a sequence:


 * $$\left\{2,4,6,8\right\}$$


 * $$2$$ is the first term in the sequence and has an ordinal position of 1.
 * $$4$$ is the second term in the sequence and has an ordinal position of 2.
 * $$6$$ is the third term in the sequence and has an ordinal position of 3.
 * $$8$$ is the fourth term in the sequence and has an ordinal position of 4.

The example sequence above has four terms and sequences with a finite number of terms are called finite sequences. However, a sequence can contain an infinite number of terms and as might be expected are called infinite sequences. Infinite sequences are of greater consideration when studying the attributes of sequences and later when studying series.

Representation
Sequences can be represented using algebraic notation, as in the example above, or on a number line or in a graph.

Using algebraic notation there are patterns for referencing the entire sequence, individual terms, describing sequences by there membership (as in the example above) or by rules composed of formulas that can include a variable containing the position of the term. The basic definition of a sequence contains the name of the sequence equated to its terms:

TODO: convergent, divergent, oscillating and alternating examples to use later in the converge/diverge discussion


 * $$\begin{align}

\left\{\right\}&=\left\{1,2,3,4,...\right\} \\ \left\{{{b_n}}\right\}&=\left\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},...\right\} \\ \left\{{{c_n}}\right\}&=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},...\right\} \\ \left\{{{d_n}}\right\}&=\left\{-1,1,-1,1...\right\} \end{align}$$

In this example $$\left\{ {a_n} \right\}$$, $$\left\{ {b_n} \right\}$$, $$\left\{ {c_n} \right\}$$ and $$\left\{ {d_n} \right\}$$ are all infinite sequences with the names $$a$$, $$b$$, $$c$$ and $$d$$ respectively. The first terms are prescribed and provide a pattern for deriving a formula for calculating successive terms. The ellipsis ($$...$$) indicate that the sequence continues infinitely.

The individual terms of the sequence can be referenced by the ordinal subscript. In the sequence $$\left\{{{c_n}}\right\}=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},...\right\}$$:



\begin{align} {c_1}&=1 \\ {c_2}&=\frac{1}{2} \\ {c_3}&=\frac{1}{3} \\ {c_4}&=\frac{1}{4} \end{align}$$

Since the ellipsis indicate an infinite sequence then the next terms follow a pattern based on the relation of the prescribed terms. For this example the next terms are:


 * $${c_5}=\frac{1}{5}$$
 * $${c_6}=\frac{1}{6}$$

where the denominator of the fraction is equal to the ordinal position of the term.

Another pattern for defining a sequence is to include a rule in the definition, for example:


 * $$\begin{align}

\left\{\right\}&=\left\{1,2,3,4,...,{n_{n-1}}+1,...\right\} \\ \left\{{{b_n}}\right\}&=\left\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},...,\frac{n}{n+1},...\right\} \\ \left\{{{c_n}}\right\}&=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{n},...\right\} \\ \left\{{{d_n}}\right\}&=\left\{-1,1,-1,1...,(-1)^n,...\right\} \end{align}$$

which can be abbreviated by removing the prescribed terms:


 * $$\begin{align}

{a_n}&={n_{n-1}}+1 \\ {b_n}&=\frac{n}{n+1} \\ {c_n}&=\frac{1}{n} \\ {d_n}&=(-1)^n \end{align}$$

By default the ordinal of the first term is "1". However, the starting ordinal can be prescribed using a subscript as in:


 * $$\left\{ {{a_n}} \right\} = \left\{ \frac{1}{n} \right\}_{n = 4}^\infty $$

will produce the sequence:


 * $$\left\{\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},...\right\}$$


 * TODO sectionNumber Lines and Graphs of Sequences

Convergence and Divergence
Successive terms in a sequence either converge to a particular number, diverge towards positive or negative infinity, or oscillate between numbers and neither approach a particular number nor approach infinity. There is a relationship to the theory of limits and the convergence and divergence of sequences.

Consider the relationship between consecutive terms in the following divergent sequence:


 * $$\left\{1,4,9,16,...,n^2,...\right\}$$

TODO: corresponding graph for sequence

Notice that each successive term is greater than the previous. That is, $${a_1}<{a_2}$$ and $${a_2}<{a_3}$$ and so on for each prescribed term. Intuitively this pattern will continue where each successive term is greater than the previous term and as the original position nears infinity then the value of the term intuitively reaches positive infinity. The sequence diverges towards positive infinity. A sequence can also diverge towards negative infinity as in:


 * $$\left\{-1,-4,-9,-16,...,-n^2,...\right\}$$

TODO: corresponding graph for sequence

where each successive term is less than the previous. That is, $${a_1}>{a_2}$$ and $${a_2}>{a_3}$$ and so on for each term.


 * $$\left\{1,\frac{1}{4},\frac{1}{9},\frac{1}{16},...,\frac{1}{n^2},...\right\}$$


 * $${\mathbb{R}_{ > 0}} = \left\{ {x \in \mathbb{R}\left| {x > 0} \right.} \right\}

$$
 * $$\varepsilon \in {\mathbb{R}_{ > 0}}$$
 * $$N \in \mathbb{Z}$$
 * $${\forall _\varepsilon }{\exists _N}{\forall _n}\left( {\left(n > N\right) \to \left(\left| {{a_n} - L} \right| < \varepsilon \right)} \right)$$

TODO definition: converges and diverges TODO definition: diverges to infinity and diverges to negative infinity

Limits
TODO theorem: sum, difference, constant multiple, product and quotient rule TODO theorem: sandwich theorem for sequences TODO theorem: continuous function theorem for sequences

L'Hopital's Rule
TODO theorem: L'Hopital's


 * Common Limits of Sequences

Recursion

Bounded Monotonic Sequences
TODO definition: unbounded, bounded, upper bound, least upper bound, lower bound, greatest lower bound TODO definition: nondecreasing, nonincreasing, monotonic TODO theorem: monotonic sequence theorem

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