find the power series representation of the function.
heres the equation: http://img260.imageshack.us/img260/9021/untitledph1.jpg
need help.
thanks.
find the power series representation of the function.
heres the equation: http://img260.imageshack.us/img260/9021/untitledph1.jpg
need help.
thanks.
We know that, $\displaystyle -1<x\leq 1$
$\displaystyle \tan^{-1}x=x-\frac{x^3}{3}+\frac{x^5}{5}-...$
Know if you multiply this term by term for $\displaystyle (1+x^2)$ we have,
$\displaystyle x-\frac{x^3}{3}+\frac{x^5}{5}-....+x^3-\frac{x^5}{3}+\frac{x^7}{5}-...$
$\displaystyle x+\frac{2x^3}{3\cdot 1}-\frac{2x^5}{5\cdot 3}+\frac{2x^7}{7\cdot 5}-\frac{2x^9}{9\cdot 7}+...$
I just need to make the restriction,
$\displaystyle -1<x<1\,$ rather then the endpoint because the Cauchy product applies only for the interval of absolute convergence.