1. ## Taylor polynomial #2

Calculate General Polynomail and determine the remainder term

g(x) = (1+x)^(1/3) , x=0

2. 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

$\displaystyle \chi$ $\displaystyle \sigma$

3. 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?

4. 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

$\displaystyle \chi$ $\displaystyle \sigma$

5. The theorem in my book says 'Taylors Remainder theorem' (which is very confuzing)

So I assume it is the taylor remainder

6. 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

$\displaystyle \chi$ $\displaystyle \sigma$