Power series representation of a function

I am trying to find a power series representation of

$$\frac{1-x}{1+x}$$

My textbook does it by adding one and subtracting one on the numerator. I understand this method but was wondering if anyone could give me some input on doing it by the following method:

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

where the series is just the geometric series replaced by $-x$.

After multiplying out I get stuck here:

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

I'm not sure how I could progress or if this method is actually doable, any input would be appreciated thanks!

• @Reveillark Did you not read all the way through the problem? – Simply Beautiful Art Feb 4 '17 at 0:29
• Yes! I understand there are other ways to do it much like the one done in my textbook, I was just hoping to get some input on the method I stated above and whether it's valid or not. – melm Feb 4 '17 at 0:34

Re-index:

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

$$=-1+2\sum_{n=0}^\infty(-1)^nx^n$$

• You haven't done the reindexing right: the constant term should be $1$ not $2$. – Rob Arthan Feb 4 '17 at 0:40
• This looks like just the answer I need! I'm just a bit confused on the second step and how you got the -x at the end? – melm Feb 4 '17 at 0:41
• @RobArthan I can't find what I did wrong. Could you explain? – Simply Beautiful Art Feb 4 '17 at 0:43
• Doesn't it look like this: $(1 - x + x^2 - x^3 + \ldots) - (x - x^2 + x^ 3 - \ldots) = 1 - 2x + 2x^2 - 2x^ 3 + \ldots$? – Rob Arthan Feb 4 '17 at 0:44
• @melm Sorry, RobArthan caught a mistake. Indeed, I think looking at his comment may do you better. Now, I added one to the second original sum and then re-indexed everything to look better, and to preserve equality, I subtracted one as well. – Simply Beautiful Art Feb 4 '17 at 0:48

I would prefer do the following \begin{align} \frac{1-x}{1+x}&=-1+\frac{2}{1+x}\\ &=-1+2\sum_{n=0}^{+\infty}(-1)^nx^n\\ &=\left[-1+\sum_{n=0}^{+\infty}(-1)^nx^n\right]+\sum_{n=0}^{+\infty}(-1)^nx^n\\ &=\left[-1+(1-x+x^2-x^3+\dots)\right]+\sum_{n=0}^{+\infty}(-1)^nx^n\\ &=(-x+x^2-x^3+\dots)+\sum_{n=0}^{+\infty}(-1)^nx^n\\ &=-x(1-x+x^2-x^3+\dots)+\sum_{n=0}^{+\infty}(-1)^nx^n\\ &=-x\sum_{n=0}^{+\infty}(-1)^nx^n+\sum_{n=0}^{+\infty}(-1)^nx^n\\ &=(-x+1)\sum_{n=0}^{+\infty}(-1)^nx^n. & \end{align}