# Taylor Series

soitgoes2019

## Homework Statement

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

## 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...

## Answers and Replies

Mentor
2022 Award
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

Mentor
2022 Award
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.

Homework Helper
Dearly Missed

## Homework Statement

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

## 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##.