Calculate General Polynomail and determine the remainder term
g(x) = (1+x)^(1/3) , x=0
Please help
It exists a general Taylor expansion which is known as 'Binomial series' ...
$\displaystyle (1+x)^{\alpha} = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^{n}$ (1)
... where...
$\displaystyle \binom{\alpha}{n} = \frac{\alpha\cdot (\alpha-1)\cdot (\alpha-2) \dots (\alpha-n+1)}{n!}$ (2)
Setting in (1) $\displaystyle \alpha=\frac{1}{3}$ we obtain...
$\displaystyle (1+x)^{\frac{1}{3}} = 1 + \frac{x}{3} -\frac{x^{2}}{9} + \frac{5\cdot x^{3}}{81} - \dots$ (3)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Thanks you!
My professor wants $\displaystyle (1+x)^{\frac{1}{3}} = 1 + \frac{x}{3} -\frac{x^{2}}{9} + \frac{5\cdot x^{3}}{81} - \dots
$ put into a formula to get rid of the ... (I don't know how to do that). Also, what is the remainder?
The Taylor expansion of a function has different 'remainders', each of them takes its name from a mathematician of ther past: for example Cauchy, Lagrange, Peano,... which is the 'remainder' that your professor requires?...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Some examples of 'remainder'...
http://mathworld.wolfram.com/LagrangeRemainder.html
http://mathworld.wolfram.com/CauchyRemainder.html
http://mathworld.wolfram.com/SchloemilchRemainder.html
... but they are not 'alone'... what is your professor's 'preference'?...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$