Power series representation question

[kvkenyon]

Homework Statement

Determine the value of f(-1) when

Homework Equations

f(x) = (x/2^2) + ((2x^3)/2^4)+((3x^5)/2^6)+... .

(Hint: differentiate the power series representation of ((x^2)-2^2)^(-1).)

The Attempt at a Solution

I was not very sure were to begin on this one. So I followed what the hint said.

the power series representation is n = 0 to infinity -sigma x^(2n)/4^(n+1)

Then i tried taking the derivative but it left me with no idea were to continue...frustrated to say the least

 

[kvkenyon]

work:

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

= -1/4 sigma ((x^2)/4)^n

= - sigma (x^2n) / 4^n+1

= 1/4 +x^2/4^2 + x^4/4^3 + x^6 / 4^4 + .....

now i try and take the derivative but i dont see what the point is

 

[kvkenyon]

please someone help

 

[LCKurtz]

Homework Statement

Determine the value of f(-1) when

Homework Equations

f(x) = (x/2^2) + ((2x^3)/2^4)+((3x^5)/2^6)+... .

(Hint: differentiate the power series representation of ((x^2)-2^2)^(-1).)

The Attempt at a Solution

I was not very sure were to begin on this one. So I followed what the hint said.

the power series representation is n = 0 to infinity -sigma x^(2n)/4^(n+1)

Then i tried taking the derivative but it left me with no idea were to continue...frustrated to say the least

OK, assuming you have the power series correct, let's call your power series representation g(x):

$$g(x) = -\sum_{n=0}^\infty \frac{x^{2n}}{4^{n+1}}$$

Differentiate it:

$$g'(x) = -\sum_{n=1}^\infty \frac{2nx^{2n-1}}{4^{n+1}} = -\sum_{n=1}^\infty \frac{2nx^{2n-1}}{2^{2n+2}}$$

Cancel out one of the 2's and factor out a 1/2 and it looks a lot like your problem.

 

[kvkenyon]

ok it is -4/9 thank you so much. Next time ill just follow orders