Why Does Differentiating a Geometric Series Lead to an Alternating Series?

[fk378]

Homework Statement

Use differentiation to find a power series representation for f(x) = 1/ (1+x)^2

Homework Equations

geometric series sum = 1/(1+x)

The Attempt at a Solution

(1) I see that the function they gave is the derivative of 1/(1+x).
(2) Therefore, (-1)*(d/dx)summation(x^n) = -1/(1+x)^2
(3) Differentiating the summation gives:
(-1)*[summation (n)x^(n-1)]

However, the book is telling me that for my second step (2) I should be getting
d/dx [summation (-1)^n (x^n)].

Why is it becoming an alternating series here?

 

[StatusX]

Remember:

$$\frac{1}{1-x} = 1+x+x^2+...$$

so that, after substituting -x for x:

$$\frac{1}{1+x} = 1 - x + x^2 + ...$$

You can remember the denominator in the first equation is 1-x by multiplying both sides by (1-x), giving:

$$1 = (1-x)(1+x+x^2+...) = 1 + x + x^2 + ... - x - x^2 -x^3 - ... = 1$$

which is consistent, as opposed to what you'd get if you assume 1+x+... was 1/(1+x). (By the way, these manipulations of infinite sums aren't strictly valid, but they can be made more rigorous by restricting to finite sums and taking a limit at the end).