Taylor Series Expansion for f(z) = −1/z^2 about z = i + 1

[Applejacks]

Homework Statement

Find the Taylor series expansions for f(z) = −1/z^2 about z = i + 1.

Homework Equations

The Attempt at a Solution

I'm just not sure what format I'm supposed to leave it in.

Is it meant too look like this:
f(z)=f(i+1)+f'(i+1)(x-i-1)...

or this
Ʃ$\frac{1}{n!}$$f^{(n)}$(1+i) * (z-i-1)^n (also is this correct?)

 

[Dick]

They are exactly the same thing. The first expression is just the first two terms of the second.

 

[Applejacks]

Yeah I'm aware of that. I guess I should keep it in the second form though. Another question: what does the square do to the function. What I wrote can't be correct because -1/z would give the same thing.

Ʃ$\frac{1}{n!}$$f^{(n)}$(1+i) * (z-i-1)^2n

 

[Dick]

Yeah I'm aware of that. I guess I should keep it in the second form though. Another question: what does the square do to the function. What I wrote can't be correct because -1/z would give the same thing.

Ʃ$\frac{1}{n!}$$f^{(n)}$(1+i) * (z-i-1)^2n

What?? Changing f changes f^(n)(i+1). That changes the series doesn't it? The power part (z-i-1)^n doesn't change. Those are the powers in the expansion of any function around i+1.

 

[Applejacks]

ah right. Thanks for pointing that out.