Thread: Remainder of a Taylor Series.

1. Remainder of a Taylor Series.

Could someone give me a quick run through on how to evaluate the Remainder term of a Taylor series?

I'm supposed to calculate the 4th Taylor polynomial of:

$\sqrt{1+x}$

I did this and got:

$T_{4}~=~\frac{x^{3}}{512}-\frac{17x^{2}}{512}+\frac{203x}{512}+\frac{541}{51 2}$

And I found that the nth derivative of this function can be given by:

$f^{\left(n\right)}\left(x\right)~=~\left(-1\right)^{\left(n-1\right)}\frac{\left(2n-3\right)!!}{2^{n}\left(x+1\right)^{\frac{2n-1}{2}}}$

I'm not 100% sure how to calculate the remainder now.

2. I made a small change, it used to say (n-5)!!

Now it is (n-3)!! as it should be.