I need help with the maclaurin series

Question

anonymous · Answer

Find the series 

f(x) = (1-x)^-2

please use lots of detail

anonymous · Answer

http://upload.wikimedia.org/wikipedia/en/math/d/5/a/d5a2af7ba7c8936d29ccb8a2b53a18ed.png

anonymous · Answer

we need to find a way to turn (1-x)^-1 into (1-x)^-2

anonymous · Answer

let's try differentiating (1-x)^-1

anonymous · Answer

ok, so i know how to get the derivative the problem is is i feel like it should be alternating but my book answer is not showing that

anonymous · Answer

it shouldn't be , since you take derivative of one side , you also take derivative of other side http://upload.wikimedia.org/wikipedia/en/math/d/5/a/d5a2af7ba7c8936d29ccb8a2b53a18ed.png

anonymous · Answer

ok that part doesnt make since to me at all by the time we get to the 4th derivative we should have -2/(1-x)^3 then 6/(1-x)^4 and it should alternate back and forth... I know im not doing something right. I'm not saying your wrong
haha

anonymous · Answer

oh it is much simpler than that


you know
$$\sum _{n=0}^{\infty } x^n=\frac{1}{1-x}$$

when we take derivative of right sides , we get

$$(1-x)^{-2}$$

which would equal derivative of left sides

$$\sum _{n=0}^{\infty } n x^{n-1}$$

anonymous · Answer

but still wouldnt the negative one come down?