Why same initial value in power series

[shanepitts]

Homework Statement

Find a power series representation for the function below & determine the radius of convergence.

f (x)=(1+x)/(1-x)²

2.Relevant equation

Shown in attached image below which is the solution the problem.

3.The attempt at a solution

I'm trying to fathom the solution here.

I am pretty sure the initial value is the value of n. If so, it does not seem that the starting values were made equal here. Unless it has something to do with the 1+ ∑ that shows up on the fourth line?

Please help

 

[Mark44]

Homework Statement

Find a power series representation for the function below & determine the radius of convergence.

f (x)=(1+x)/(1-x)²

2.Relevant equation

Shown in attached image below which is the solution the problem.

3.The attempt at a solution

View attachment 84215
I'm trying to fathom the solution here.

I am pretty sure the initial value is the value of n. If so, it does not seem that the starting values were made equal here. Unless it has something to do with the 1+ ∑ that shows up on the fourth line?

The starting values are not equal in the third line. The comment to this effect is not that the starting values were made equal, but rather that this is the goal, which happens in the 4th line.

Note that $\sum_{n = 0}^{\infty} (n + 1)x^n = 1x^0 + \sum_{n = 1}^{\infty} (n + 1)x^n = 1 + \sum_{n = 1}^{\infty} (n + 1)x^n$. Is that what you're asking about?

 

[shanepitts]

The starting values are not equal in the third line. The comment to this effect is not that the starting values were made equal, but rather that this is the goal, which happens in the 4th line.

Note that $\sum_{n = 0}^{\infty} (n + 1)x^n = 1x^0 + \sum_{n = 1}^{\infty} (n + 1)x^n = 1 + \sum_{n = 1}^{\infty} (n + 1)x^n$. Is that what you're asking about?

Thanks for your quick response. I fathom now.