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?