coefficients of a power series

Question

I have the function $$f(x)=\frac{2x}{10+x}$$ and I am asked to find its power series representation which I found to be $$\sum_{n=0}^{\infty} (-1)^{n} *\frac{2x^{n+1}}{10^{n+1}}$$ and I found the radius of convergence to be $R=10$. All until here is clear and easy, but when I am asked to find the 1st few terms I tries to do the following

• $c_0$ I plugged a value of $x=0$ in my original function $f(x)$ which equals $(0)$ [correct answer]

• $c_1$ I plugged a value of $x=0$ in the 1st derivative of $f(x)$ which equals $\frac{1}{5}$ [correct answer]

• $c_2$ I plugged a value of $x=0$ in the 2nd derivative of $f(x)$ [incorrect answer]

• $c_3$ I plugged a value of $x=0$ in the 3rd derivative of $f(x)$ [incorrect answer]

• $c_4$ I plugged a value of $x=0$ in the 4th derivative of $f(x)$ [incorrect answer]

So if $c_0$ and $c_1$ are correct why would the others not be as well? Am I missing something profoundly important?

Answer 1

$$\dfrac{2x}{10+x}=\dfrac{2x}{10\left(1+\dfrac x{10}\right)}=\dfrac x5\left(1+\dfrac x{10}\right)^{-1}$$

For $\left|\dfrac x{10}\right|<1,$ $$\left(1+\dfrac x{10}\right)^{-1}=\sum_{r=0}^\infty\left(-\dfrac x{10}\right)^r$$