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37 lines
1.1 KiB
Markdown
37 lines
1.1 KiB
Markdown
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title: "taylor"
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---
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#math/calculus
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# formula
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#card/reverse
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$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n !}(x-a)^{n}$
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$n ! \quad=$ factorial of $\mathrm{n}$
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$a \quad=$ real or complex number
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$f^{(n)}(a)=$ nth derivative of $f$ evaluated at the point a
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![[Pasted image 20220603201244.png]]
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mclaurin series is the same, except a is 0
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# When it's a power series
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if $\sum\limits_{n=0}^{\infty}c_{n}(x-a)^{n}= f(x)$,
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so
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if $f(x)=\lim_{x->\infty}T(x)$ and $T(x)=\sum\limits_{i=0}^{n}f^{i}\frac{a}{i!}(x-a)^i$, and remainder $Rn(x)=f(x)-T(x)$ its a power series if $lim_{x->0}Rn(x)=0$
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$|R_{n}(x)|\leq \frac{M}{(n+1)!}|x-a|^{n+1}$
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if $|f^{n+1}(x)|\leq M|x-2|\leq d$
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for example, in the case of a taylor series for $e^{x}$ centered at 2: $|e^{x}|\leq e^{d+2}= M$ thus $|Rn(x)|\leq \frac{e^{d+2}}{(n+1)!}d^{n+1}$ lim both approach zero, so $e^{x}= \sum\limits_{n=0}^{\infty} \frac{e^{2}}{n!}(x-2)^n$
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# nice power series
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$e^n=$ ::: $\sum\limits \frac{x^{n}}{n!}$
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$\frac{1}{1-x}=$ ::: $\sum\limits x^{n}$
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$sin(x)$ ::: $\sum\limits \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}$
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$arctan(x)$ ::: $\sum\limits \frac{(-1)^{n}x^{2n}}{2n!}$
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[[binomial series]] |