1.1 KiB
| title |
|---|
| taylor |
#math/calculus
formula
#card/reverse
$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n !}(x-a)^{n}$
n ! \quad= factorial of $\mathrm{n}$
a \quad= real or complex number
f^{(n)}(a)= nth derivative of f evaluated at the point a
!
mclaurin series is the same, except a is 0
When it's a power series
if \sum\limits_{n=0}^{\infty}c_{n}(x-a)^{n}= f(x),
so
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
$|R_{n}(x)|\leq \frac{M}{(n+1)!}|x-a|^{n+1}$
if |f^{n+1}(x)|\leq M|x-2|\leq d
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
nice power series
e^n= ::: $\sum\limits \frac{x^{n}}{n!}$
\frac{1}{1-x}= ::: $\sum\limits x^{n}$
sin(x) ::: $\sum\limits \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}$
arctan(x) ::: \sum\limits \frac{(-1)^{n}x^{2n}}{2n!}