Question

How do you find a power series representation for `(1+x)/((1-x)^2)`?

Answer 1

Fair warning---I expect this to be a long answer!

I got

`sum_(n=0)^(N) (1+x)/(1-x)^2 = 1 + 3x + 5x^2 + 7x^3 + ...`

for some large finite `N`.

---

So, something that I believe you have already been taught is that:

`1/(1-x) = sum_(n = 0)^N x^n = 1 + x + x^2 + x^3 + ...` (0)

for some large finite `N`. Notice how:

`d/(dx)[-1/(1-x)] = 1/(1-x)^2` (1)

and that:

`color(darkred)(-1/(1-x)) = -sum_(n = 0)^N x^n` (2)

Analogizing from (1):

`x/(1-x)^2 = x*["Power Series for "1/(1-x)^2]` (3)

Remember these four relationships, because we will be referring back to them.

First, notice how you can rewrite this as:

`(1+x)/(1-x)^2 = color(green)(1/(1-x)^2 + color(red)(x/(1-x)^2))` (4)

Already you may see how things could unfold. Making use of (0) in conjunction with (2), we get:

`color(darkred)(-1/(1-x) = -1 - x - x^2 - x^3 - ...)`

Now, making use of (1), we get:

`color(green)(1/(1-x)^2) = d/(dx)[-1 - x - x^2 - x^3 - ...]`

`= color(green)(-1 - 2x - 3x^2 - ...)`

This becomes the left half that we are looking for. Now for the right half. Using (3), we get:

`color(red)(x/(1-x)^2) = x[-1 - 2x - 3x^2 - 4x^3 - ...]`

`= color(red)(-x - 2x^2 - 3x^3 - ...)`

Next, we can add them together (like-terms with like-terms) according to (4):

`"Power Series of " (1+x)/(1-x)^2 = "Power Series of " color(green)(1/(1-x)^2 + color(red)(x/(1-x)^2))`

`prop [-1 - 2x - 3x^2 - 4x^3 - ...] + [-x - 2x^2 - 3x^3 - ...]`

`prop color(green)(-1 - 3x - 5x^2 - 7x^3 - ..).`

And finally, notice how we started with `1/(1-x)`, but we manipulated it before taking the derivative to get `1/(1-x)^2` by multiplying it by `-1` to get the power series for `-1/(1-x)`.

The negative sign carried through all our operations.

Therefore, to get the true power series, we have to undo the multiplication we did on `1/(1-x)` by dividing by `-1` at the end, thus dividing the previous result by `-1`. So, finally, we get:

`color(blue)(sum_(n=0)^(N) (1+x)/(1-x)^2 = 1 + 3x + 5x^2 + 7x^3 + ...)`

(that is why I purposefully put the `prop` symbol to emphasize that we were not at the final answer yet)

Wolfram Alpha agrees with this answer.