This is going to be pretty hard to show as text since it would be easier for me to post a picture of the question.

Question

The question has f(x) = x/4^2 - (2x^3)/4^4 + (3x^5)/4^6 + ... .

I am trying to find out the value of f(2). There is a hint to differentiate the power series expansion of 1/(x^2 + 4^2).

I've found that f(x) can be written as a series of (sigma) (((-1)^n)(n)(x^(2n-1))) / (4^(2n))

The picture has the question that I'm trying to figure out.

When I tried to do the derivative of the power series expansion of (1/(x^2+16)), I didn't really come up with anything useful. What I noticed was that the derivative of the power series expansion ended up to be f(x) times -2/16, I'm not sure if this even helps me go to the right direction.