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46 lines
1019 B
Markdown
46 lines
1019 B
Markdown
---
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cards-deck: default_obsidian
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---
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#math/calculus
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# definition of series
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#card/reverse
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The sum of a [[sequence]]. Use the $\Sigma_{n=1}^\infty$ notation.
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^1652969721739
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# partial sum
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It's a series that converges to a value. So, you're doing *part* of the sum at a time.
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# algebraic vs geometric
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Algebraic series ::: the function uses n in the function as a normal coefficient. ^1652972454834
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# Geometric series
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#card/reverse
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the function in the series uses n as an exponent. Thus, the growth is big very cool.
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^1652972485160
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$a$ is the constant, $r$ is rate
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Formula for value at any n: ==$S_n=\frac{a(1-r^{n})}{1-r}$==
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^1652972574160
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Formula for value at infinity(a is the start): ==$S_n=\frac{a}{1-r}$==
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^1653167523226
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## nice little formula
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$\Sigma_{n=1}x^{n}$ and $|x|<1$
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$a=x$
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$= \frac{x}{1-x}$
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if:
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$n=0$
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$a=1$
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$= \frac{1}{1-x}$
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# sum properties
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- $\Sigma c\cdot a_{n} = c \cdot \Sigma a_{n}$
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- $\Sigma (a_{n}+b_{n}) = \Sigma a_{n}+ \Sigma b_n$
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- $\Sigma (a_{n}-b_{n}) = \Sigma a_{n} - \Sigma b_n$
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