Taylor Series Expansion to Compute Derivatives

[trust]

Homework Statement

Find the Taylor series expansion of

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

about x=1 and use this to compute f⁽⁹⁾(1) and f⁽¹⁰⁾(1)

Homework Equations

The sum from n=0 to infinity of f^(k)(c)/(k!) (x-c)^k

The Attempt at a Solution

I'm not sure how to approach this problem. Using the expansion formula is clearly incorrect as the derivative keeps on expanding. Any help would be greatly appreciated.

 

[HallsofIvy]

First, simplify by letting u= x- 1. Now the function is $u/(1- u^2)$.

The sum of the geometric series $\sum_{n=0}^\infty ar^n$ is a/(1- r) so let a= u and $u= r^2$

 

[trust]

I'm a little confused. I believe the bottom would become 1+u².

I tried using the power series for 1/(1-x) (sum from n = 0 to infinity of x^n).

This left me with the sum from n = 0 to infinity of (x-1)^(2n+1).

I then wrote out the Taylor Polynomial until the 9th power because I wanted to evaluate f⁽⁹⁾(1).

When I plug in 1, all terms go to zero. I don't think this is correct.

Any thoughts?

 

[trust]

Please find my attempt at the solution attached

 

[vela]

You're fine up to
$$\frac{u}{1+u^2} = u \sum_{n=0}^\infty (-u^2)^n$$You want to pull the sign out first before pulling the factor of u from out front into the series.
$$\frac{u}{1+u^2} = u \sum_{n=0}^\infty (-1)^n(u^2)^n = \sum_{n=0}^\infty (-1)^n u^{2n+1} = \sum_{n=0}^\infty (-1)^n (x-1)^{2n+1}$$
Now compare this series to the Taylor expansion about x=1:
$$f(x) = \sum_{m=0}^\infty \frac{f^{(m)}(1)}{m!} (x-1)^m$$You should be able to read off what the 9th and 10th derivatives are simply by looking at the series you obtained above.

 

[trust]

I'm sorry, could you please elaborate further on how to find the derivatives?

I have attached my attempt but I don't think it's correct because I'm getting 0 for both the 9th and 10th derivatives.

 

[vela]

Like I said, compare the series you got with the generic series for the Taylor expansion.

 

[trust]

I think I am missing something.

Do I actually have to compute the 9th derivative of the original function?

Another attempt is attached.

 

[trust]

Can someone please help me with this? It is very important.