1. ## power serie representation

I 've got huge problem with one more example. Didi anyone know hoe to find power series representation of the function:

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

Thank you all for all posible answeres

2. Originally Posted by mauro21pl
I 've got huge problem with one more example. Didi anyone know hoe to find power series representation of the function:

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

Thank you all for all posible answeres
What is the integral of f(x)?
(Hint: it is an arctangent).

Thus,
INTEGRAL f(x) dx= -atan(4x)
Now, express,
-atan(4x)
As an arctangent series.

Afterwards take the derivative term-by-term.

3. Originally Posted by mauro21pl
I 've got huge problem with one more example. Didi anyone know hoe to find power series representation of the function:

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

Thank you all for all posible answeres
Use the binomial theorem on 1/(1+2(x)^2)^2:

-4x/ (1+2(x)^2)^2=-4x{1+(-2)(2x^2)+(-2)(-3)/2! (2x^2)^2+...}

.........................=sum(n=0, inf) (n+1) (-1)^(n+1) 2^(n+2) x^(2n+1)

RonL

### integralf(x)=4X THE POWER OF3

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