Using Maclaurin series to find 2005-order derivative

[hqjb]

Homework Statement

Let f(x) = \arctan(\frac{1+x}{1-x})
Find f^{2005}(0)

Homework Equations

I'm guessing this has to do with maclaurin's?

The Attempt at a Solution

...
<br /> f(x) = \pi /4 + \sum^∞_{n = 0} \frac{(-1)^n}{2n+1}x^{2n+1}<br />

\sum^∞_{n = 0}\frac{f^n(0)x^n}{n!} = \pi /4 + \sum^∞_{n = 0} \frac{(-1)^n}{2n+1}x^{2n+1}

So anyone knows how I go about from here? The answer is 2004!(factorial)
How do you compare two infinite series can you cancel them out?

 

[Harrisonized]

(d/dx)xⁿ = nx^n-1

(d^p/dx^p) xⁿ = n!/(n-p)! x^n-p
(can you see where this formula comes from?)

Also, (d/dx)C=0

They want the value of the 2005th derivative at 0. If you think about it, the term involving x²⁰⁰⁵ will become a constant after 2005 differentiations. All terms before x²⁰⁰⁵ will disappear due to (d/dx)C=0, and all terms after are irrelevant since we are calculating at 0.

The term at x²⁰⁰⁵ is supplied by the formula you found:

Ʃ (-1)ⁿ/(2n+1) x²ⁿ⁺¹

Of course, the π/4 term also disappears.

 

[lol_nl]

You can compare the two series since they are polynomials in x, and what you are interested in is only one of the terms of the polynomial. f^(2000) occurs exactly in the term of x^(2005), so you can ignore all other terms and compare the coefficients of x^2005 on both sides.

 

[hqjb]

(d/dx)xⁿ = nx^n-1

(d^p/dx^p) xⁿ = n!/(n-p)! x^n-p
(can you see where this formula comes from?)

Also, (d/dx)C=0

They want the value of the 2005th derivative at 0. If you think about it, the term involving x²⁰⁰⁵ will become a constant after 2005 differentiations. All terms before x²⁰⁰⁵ will disappear due to (d/dx)C=0, and all terms after are irrelevant since we are calculating at 0.

The term at x²⁰⁰⁵ is supplied by the formula you found:

Ʃ (-1)ⁿ/(2n+1) x²ⁿ⁺¹

Of course, the π/4 term also disappears.

You can compare the two series since they are polynomials in x, and what you are interested in is only one of the terms of the polynomial. f^(2000) occurs exactly in the term of x^(2005), so you can ignore all other terms and compare the coefficients of x^2005 on both sides.

Ah...i get it now so,
f^{2005}(0)=\frac{2005!}{2005} * (-1)^{1002}

Thanks for your help!