Taylor series centered at c = 1

[DrummingAtom]

Homework Statement

Find the Taylor Series of 1/x centered at c = 1.

Homework Equations

$$\sum_{n=0}^{\infty} f^n (c) \frac{(x-c)^n}{n!}$$

The Attempt at a Solution

I made a list of the derivatives:

f(x) = 1/x
f'(x) = -1/x²
f''(x) = 2/x³
f'''(x) = -6/x⁴

f(1) = 1
f'(1) = -1
f''(1) = 2
f'''(1) = -6

From this I see the pattern fⁿ(c) = (-1)ⁿ(n!)

$$\sum_{n=0}^{\infty} (-1)^n(n!) \frac{(x-1)^n}{n!}$$

Then I canceled the factorials and I'm left with

$$\sum_{n=0}^{\infty} (-1)^n (x-1)^n$$

Checked my answer and it's way off.. Thanks for any help.

 

[Dick]

Homework Statement

Find the Taylor Series of 1/x centered at c = 1.

Homework Equations

$$\sum_{n=0}^{\infty} f^n (c) \frac{(x-c)^n}{n!}$$

The Attempt at a Solution

I made a list of the derivatives:

f(x) = 1/x
f'(x) = -1/x²
f''(x) = 2/x³
f'''(x) = -6/x⁴

f(1) = 1
f'(1) = -1
f''(1) = 2
f'''(1) = -6

From this I see the pattern fⁿ(c) = (-1)ⁿ(n!)

$$\sum_{n=0}^{\infty} (-1)^n(n!) \frac{(x-1)^n}{n!}$$

Then I canceled the factorials and I'm left with

$$\sum_{n=0}^{\infty} (-1)^n (x-1)^n$$

Checked my answer and it's way off.. Thanks for any help.

How did you check your answer to conclude "it's way off"? I agree with your answer. I'll probably disagree with your "check".

 

[DrummingAtom]

Hmm well that's reassuring. We've had a couple problems with the answers in this book being off.

The back of the book said the answer is:

$$\sum_{n=0}^{\infty} \frac {(-1)^{n+1}(x-5)^n} {4^{n+1}}$$

I'll just go ahead and ignore that answer for now until next class. Thanks for your help.

 

[Dick]

Hmm well that's reassuring. We've had a couple problems with the answers in this book being off.

The back of the book said the answer is:

$$\sum_{n=0}^{\infty} \frac {(-1)^{n+1}(x-5)^n} {4^{n+1}}$$

I'll just go ahead and ignore that answer for now until next class. Thanks for your help.

The book's answer doesn't even converge at x=1 and it's not expressed in powers of (x-1). It looks like the answer to some completely different exercise. Yes, ignore it.