25 lines
1.7 KiB
Markdown
25 lines
1.7 KiB
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. $\frac{a_n}{b_n} \to \frac{L}{M}$ holds true where all values are defined
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5. If $L = M$ and a sequence $c_n$ exists such that $a_n \le c_n \le b_n$ for all $n$, then $c_n \to L = M$
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6. If $a_n$ and $b_n$ both approach infinity at a similar rate, $\frac{a_n}{b_n}$ will approach an arbitrary value. This value can be found by rewriting $\frac{a_n}{b_n}$ in such a manner that the end behavior of the series is more easily identifiable
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> For example, given the series $c_n = \frac{n}{2n+1}$, both the numerator and the denominator approach infinity at a similar rate. However, when the numerator and denominator are both multiplied by $\frac{1}{n}$, it becomes $\frac{1}{2+\frac{1}{n}}$, an equivalent sequence that more clearly converges on $1/2$.
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