Question

Use a known maclaurin series to obtain the maclaurin series for the given function. f(x = 8x2 tan?1(7x3

Answer 1

Going out on a limb here and guessing that the function is

$f(x)=8x^2\tan^{-1}(7x^3)$

Please correct me if this isn't the case.

Recall that

$\tan^{-1}x=\displaystyle\sum_{n\ge0}\frac{(-1)^nx^{2n+1}}{2n+1}$

which converges for $|x|<1$.

It follows that

$8x^2\tan^{-1}(7x^3)=8x^2\displaystyle\sum_{n\ge0}\frac{(-1)^n(7x^3)^{2n+1}}{2n+1}$
$=\displaystyle\sum_{n\ge0}\frac{56(-49)^nx^{6n+3}}{2n+1}$