Find the first 5 coefficients of the series $\frac{6x}{x+9} = \sum_{n=0}^\infty C_n x^n$

Question

I rewrote the equation series as $$ \frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1} $$

And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) \left(\frac{-1}{9}\right)^1, C_2 = ( 6/9 )\left(\frac{-1}{9}\right)^2, C_3 = \cdots $ This is wrong however and I'm not sure why. If anyone knows the solution to this could they point me in the right direction?

Answer 1

$\frac{6x}{x+9}=6-\frac{54}{9+x}=6-\frac{6}{1-\frac{-x}{9}}=6-6\sum_{n=0}^{\infty}(-\frac{x}{9})^n \quad \forall x,\mid{x}\mid<9$.

Now let $n=0,1,2,4$ and calculate $c_{n}$s.

Answer 2

For sure, the fastest way is to build the Taylor series of the lhs and identify the coefficients.

You could also write $$6x =(x+9) \sum_{n=0}^4 C_n x^n=9 C_0+(C_0+9 C_1) x+(C_1+9 C_2) x^2+(C_2+9 C_3) x^3+(C_3+9 C_4) x^4+C_4 x^5+ ...$$ and identify the coefficients for each power of $x$. This gives $C_0=0$, $9C_1=6$ and so on.