1. ## Taylor Polynomial

Find T(10) (x): the Taylor polynomial of degree 10 of the function f(x)=arctan(x^3) at a=0.

Can someone show me the steps to finding this, please?

2. Originally Posted by Del
Find T(10) (x): the Taylor polynomial of degree 10 of the function f(x)=arctan(x^3) at a=0.

Can someone show me the steps to finding this, please?
note that if

$\displaystyle f(x)=\tan^{-1}(x)$ then

$\displaystyle f'(x)=\frac{1}{1+x^2}$

by the def of a geometric series we get...

$\displaystyle f'(x)=\sum_{n=0}^{\infty}(-1)^nx^{2n}$ so if integrate both sides we get

$\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{2n+1}$

so then $\displaystyle f(x^3)=\tan^{-1}(x^3)=\sum_{n=0}^{\infty}\frac{(-1)^n(x^3)^{2n+1}}{2n+1}$

so we get...
$\displaystyle tan^{-1}(x^3)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{6n+3}}{2n+1}$

Just write out as many terms as you need.

3. I got this massive polynomial for T(10), but it's incorrect. What am I doing wrong?

T(10) = [x^3-x^9/3+x^15/5-x^21/7+x^27/9-x^33/11+x^39/13-x^45/15+x^51/17-x^57/19]

4. Originally Posted by Del
I got this massive polynomial for T(10), but it's incorrect. What am I doing wrong?

T(10) = [x^3-x^9/3+x^15/5-x^21/7+x^27/9-x^33/11+x^39/13-x^45/15+x^51/17-x^57/19]
When they say $\displaystyle T_{10}(x)$ They mean a polynomial with degree 10 or less.

You have a polynomial of degree 57. You would want only the first two terms.

$\displaystyle T_{10}(x)=x^3-\frac{1}{3}x^9$ the next term would be degree 15

Good luck.

B