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32 lines
1.2 KiB
Markdown
32 lines
1.2 KiB
Markdown
#math/calculus
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# what it is
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[Power series - Wikipedia](https://en.wikipedia.org/wiki/Power_series)
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## power series
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#card/reverse
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it's actually a type of (simpler) [[taylor]] series. weird but true
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$\sum_{n=0}^{\infty} a_{n}(x-c)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\ldots$
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## mclaurin
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$\sum_{n=0}^{\infty} a_{n} x^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots$
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# 3 possibilities for the value of x
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1. $x=a$ [[Ratio test]] gives $\infty$
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2. $(-\infty,\infty)$, [[Ratio test]] $= 0$
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3. find $R>0$ (radius of convergence) where $|x-a|<R$ converges. [[Ratio test]] gives R where $-R+a<x<R+a$
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# intervals and derivatives
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$f'(x)= c_{1}+2c_{2}(x-a)+3c_{3}(x-a)^2...=\sum\limits_{n=1}^{\infty}nc_n(x-a)^{n-1}=\sum\limits_{n=0}^{\infty}(n+1)c_{n}(x-a)^{n}$
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$\int f(x)dx=C + c_{o}(x-a)+c_1\frac{(x-a)^2}{2}+c_2\frac{(x-a)^3}{3}...=C+\sum\limits_{n=0}^{\infty}c_n\frac{(x-a)^{n+1}}{n+1}$
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# use to solve integrals
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- any part of the integral that might be approximated by a power series can just be swapped out by its power series
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- integrate and sum operations can be swapped around
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- power series of a function is a geometric series, so its sum can be swapped in. $\sum\limits_{n=0}^{\infty}a_{n}(x-a_{n})^{n}=a\frac{1-r^{n}}{1-r}$ |