Expanding f(x) = x/(x+1) about a=10

[forestmine]

Homework Statement

Expand f(x) = x/(x+1) in a taylor series about a=10.

Homework Equations

f(x) = Ʃ (f^n(a)*(x-a)^n / n!

The Attempt at a Solution

I'm having a hard time arriving at the correct answer..I think I'm definitely getting lost somewhere along the way. Here's what I've got so far:


I started by computing the derivatives.
f'(x) = 1/(x+1)^2

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

Then evaluating each at 10:

f'(10) = 1/121

f''(10) = -22/14641

and f(10) = 10/11

Then, using the above equation,

10/11 + 1/121 * (x-10) + (-22/11^4 * (x-10)^2)/2 + ...

This doesn't really take me in the right direction at all, though. I know x/1-x is near the form 1/1-x which I need for a power series expansion. Should I be trying to represent it as such?

Hope this is clear! I'm quite confused!

 

[micromass]

Do you know the series expansion of

$$\frac{1}{1-y}$$

??

Try to write your function in that form.

 

[forestmine]

Ok, I had a feeling that was the direction I should be heading in. So, writing it in that form, I get

1/1 - (-1/x)

and so my series becomes from n=0 to infinity, (-1/x)^n

Am I on the right track at least?

 

[micromass]

Maybe I'll do an instructive example:

$$\frac{1}{x+10}=-\frac{1}{-10-x}=-\frac{1}{-18-(x-8)}=\frac{1}{18}\frac{1}{1-(x-8)/(-18)}=\frac{1}{18}\sum \frac{1}{(-18)^n}(x-8)^n$$

I hope I didn't make any typos. But that's basically it.

 

[forestmine]

Ok, I follow all of that except where you factor out 1/18. Shouldn't the 1 in the denominator, 1-(x-8)/(-18) be negative?

And then your series emerges from the fact that 1/1-x = summation x^n, where -(x-8)/-18 is your x, correct?

So, that's kinda what I tried...here it is step by step...

x/1+x = 1/(1/x+1) = 1/(1- (-1/x)

And my x for the series is (-1/x)

 

[micromass]

Ok, I follow all of that except where you factor out 1/18. Shouldn't the 1 in the denominator, 1-(x-8)/(-18) be negative?

And then your series emerges from the fact that 1/1-x = summation x^n, where -(x-8)/-18 is your x, correct?

So, that's kinda what I tried...here it is step by step...

x/1+x = 1/(1/x+1) = 1/(1- (-1/x)

And my x for the series is (-1/x)

OK, I might have made some mistakes in my post, but you get the point I see.

But what you did is NOT allowed at all. You did $\frac{1}{x+1}=\frac{1}{x}+1$ which is very, very, very wrong!

What you should do is forget the x in the numerator for a second and try to make something out of

$$\frac{1}{1+x}$$

We'll worry about the numerator later.

 

[forestmine]

Hm, I'm confused. I just factored an x out of the denominator, and canceled it with the numerator, but that's not ok?

Ok, but ignoring the numerator for now...it would just be

1/1-(-x), and so for my series, x = -x ?Thanks for the help by the way! :) I really appreciate it.

 

[micromass]

Hm, I'm confused. I just factored an x out of the denominator, and canceled it with the numerator, but that's not ok?

Your second step is not ok.

Ok, but ignoring the numerator for now...it would just be

1/1-(-x), and so for my series, x = -x ?

Remember that you would like to expand the series around a=10.
What you're doing now is fine, except that your expansion is around a=0 now.

 

[forestmine]

Ahh! That's right!

So in that case,

Ʃ(-x-10)ⁿ

 

[micromass]

No, that's not correct.

 

[forestmine]

Ah, ok, in that case, I'm lost. For taylor series, the derivative is involved, but I don't see what I'd be taking the derivative of? And I know that Taylor series is summation of f'(a)(x-a)/n!, but I'm not sure how to get this into those terms.

Edit// I take that back...I don't think the derivative is involved at all, and this is just a straight-forward power-series. Although, I'm still not sure how to represent it in terms of a. Does my a affect the original function immediately, or do I incorporate it as part of the series later?

 

[forestmine]

I've been able to make sense of my teacher's solution up to a point...

He starts with,

((x-10)+10)/((x-10) + 11)

Now, I can see how in doing that, he hasn't changed the original problem at all, and he has incorporated the a=10 right away. But...I don't really see how exactly the a is being incorporated, other than arbitrarily being thrown into the function? It could just as easily have been (x-5 + 5)/(x-5+6) for that matter, right?

From there, he breaks the above up into a sum of two fractions, and then sets (x-10) = u. And from there he proceeds to set up a series, but I don't even understand the aforementioned step, so I can't quite figure out what's going on with the series.