Finding this power series -- Where is my error?

[archaic]

Homework Statement

Find the power series representation of $$\frac{x}{(1+13x^2)^2}$$

Relevant Equations

N/A

I have used $\sim$ but meant $\sum_{k=0}^\infty$

my math homework platform is telling me that this is wrong. I've tried using desmos to test it and it was a perfect match. Any ideas on where I went wrong?

 

[Orodruin]

What type of input does the platform require? It is possible that it has written it on a different form and only accepts that form.

 

[archaic]

What type of input does the platform require? It is possible that it has written it on a different form and only accepts that form.

This is what they are giving me. If you put $k=k'+1$ in my result, then you'd get theirs, but then the sum should start from $-1$. They have specifically asked me to have it start from $0$, though, as is shown.
I've used power series instead of differentiating $f(x)$ because it is faster. The result should still be the same normally. I remember our professor saying something about trivial functions having unique taylor polynomials.

 

[Orodruin]

View attachment 260819
This is what they are giving me. If you put $k=k'+1$ in my result, then you'd get theirs, but then the sum should start from $-1$. They have specifically asked me to have it start from $0$, though, as is shown.

Your first term is equal to zero (it has a factor $k = 0$ in it). So even if you put $k = k'+1$ you can have your sum start at zero.

 

[archaic]

I've emailed my professor, maybe he'll fix it. Thank you @Orodruin I've missed the $k$ factor ^^'.