Power Series Representation of (1+x)/(1-x)

[Desharnais]

Homework Statement

For the power series representation of, f(x)=1+x1−x which is 1+2∑from n=1 to inf (x^n), Where does the added 1 in front come from? How do I get to this answer from ∑n=0 to inf (x^n)+∑n=0 to inf (x^(n+1))

Homework Equations

The Attempt at a Solution

I arrived at ∑n=0 to inf x^n + ∑ n=0 to inf x^(n+1)

 

[Kreizhn]

The 1 in front comes from using long division to isolate the 1/(1-x) term. I'm not sure what you're asking in your second question though. If you want to change the index of a summation, you can do it entirely artificially. Namely, if you want $\sum_{n=1}^\infty x^n$ to look like a sum where the lower index starts at 0 instead of 1, define a new index k = n-1.

 

[vela]

$x^0=1$

 

[Kreizhn]

Rereading your question, I now understand what you are saying. Have you been doing as suggested? Use long division to break up the rational function, then use vela's comment about how $x^0 = 1$ and you'll get the answer your originally posted.