Find Taylor Series for 1/x Around x=3

[soitgoes2019]

Homework Statement

Find the Taylor Series for f(x)=1/x about a center of 3.

Homework Equations

The Attempt at a Solution

f'(x)=-x^-2
f''(x)=2x^-3
f'''(x)=-6x^-4
f''''(x)=24x^-5
...
f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n
I'm not sure where I went wrong...

 

[fresh_42]

How did you write the sum, i.e. the requested Taylor series? And what is wrong or why do you think it is wrong?

 

[soitgoes2019]

I wrote the sum from n=0 to ∞ as: ∑-1^n (x)^-(n+1) (x-3)^n
I'm not sure if that is correctly centered at 3

 

[fresh_42]

As far as I can see, there is only a minor error; however crucial to the usage of Taylor series. You should check the general formula again.

Edit: And your formula for $f^{(n)}$ is wrong. You must not drop the coefficients all of a sudden, only because they might cancel out later in the calculation.

 

[Ray Vickson]

Homework Statement

Find the Taylor Series for f(x)=1/x about a center of 3.

Homework Equations

The Attempt at a Solution

f'(x)=-x^-2
f''(x)=2x^-3
f'''(x)=-6x^-4
f''''(x)=24x^-5
...
f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n
I'm not sure where I went wrong...

When you expand $f(x)$ about $x = 3$ your coefficients involve $f^{(n)}(3)$, not $f^{(n)}(x)$. But, of course, you compute $f^{(n)}(3)$ by first computing $f^{(n)}(x)$ and then setting $x = 3$.