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.

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.

To find the value of f(2), we can use the power series expansion of f(x) and substitute x = 2. Since f(x) is given as a power series, we can express it as a sum of terms.

The power series expansion of 1/(x^2 + 4^2) can be written as:

1/(x^2 + 4^2) = a0 + a1x + a2x^2 + ...

To find the coefficients ai, we can differentiate both sides of this equation with respect to x. Differentiating the power series term by term, we get:

-(2x)/(x^2 + 4^2)^2 = a1 + 2a2x + 3a3x^2 + ...

Comparing coefficients, we can see that a0 = 1 and a1 = 0. Therefore, the power series expansion of 1/(x^2 + 4^2) becomes:

1/(x^2 + 4^2) = 1 + 0x + a2x^2 + a3x^3 + ...

Now, let's substitute this power series expansion into the expression for f(x):

f(x) = ((-1)^n)(n)(x^(2n-1))) / (4^(2n))
= ((-1)^1)(1)(x^(2-1))) / (4^(2*1)) + ((-1)^2)(2)(x^(4-1))) / (4^(2*2)) + ...

If we substitute x = 2 into this series expression, we get:

f(2) = ((-1)^1)(1)(2^(2-1))) / (4^(2*1)) + ((-1)^2)(2)(2^(4-1))) / (4^(2*2)) + ...

Now, let's simplify this expression to find f(2).