22 lines
1014 B
Markdown
22 lines
1014 B
Markdown
A sequence is defined as an ordered list of numbers.
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- Sequences are ordered, meaning two sequences that contain the same values but in a different order are not equal.
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- Sequences can be infinite if a rule is defined, i.e $\{1, 1, 1, 1, ...\}; a_i = 1$
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# Behavior
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- A sequence is considered **increasing** if $a_n$ is smaller than $a_{n+1}$ for all $n$.
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- A sequence is considered **decreasing** if $a_n$ is greater than or equal to $a_{n+1}$ for all $n$.
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- Sequences exist that do not fall into either category, i.e, $a_n = (-1)^n$
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- If the terms of a sequence grow $\{a_n\}$ get arbitrarily close to a single number $L$ as $n$ grows larger, this is noted by writing:
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$$\lim_{n\to\infty} a_n = L$$ OR
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$$ a_n \to L \text{ as } n \to \infty $$
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and say that $a_n$ *converges* to $L$. If no $L$ exists, we say $\{a_n\}$ *diverges*.
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# Properties of Sequences
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> The below properties assume two sequences are defined, $a_n \to L$ and $b_n \to M$
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1. $a_n + b_n \to L + M$
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2. $C*a_n \to CL$
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3. $a_n b_n \to LM$
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4.
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