Thread: find the power series representation and the inteveral of convergence

1. find the power series representation and the inteveral of convergence

f(x) = (2+x)/(1-x)

I got it to....
2 sigma x^n + sigma x^(n+1)
sigma goes from 0 to infinity

I'm looking at the solution and they get...
2 + Sigma 2(x^n) + Sigma x^n = 2 + (3)Sigma x^n
In this part Sigma goes from n=1 to infinity

Can someone explain how they got to this step?

P.S. I am so sorry about not putting into the correct type format.

2. f( x ) = ( 2 + x )/( 1 - x ) = 2/( 1 - x ) + x/( 1 - x ) =

= 2*sigma( x^n ) + x*sigma( x^n ) // n goes from 0 to infinity

= 2*( 1 + x + x^2 + ... ) + x*( 1 + x + x^2 + ... ) =

= 2 + 2*( x + x^2 + x^3 + ...) + ( x + x^2 + x^3 + ... ) =

= 2 + ( 2 + 1 )*( x + x^2+ x^3 + ... ) =

= 2 + 3*sigma( x^n ) // n goes from 1 to infinity

3. This a simple geometric series.
$\frac{{2 + x}}{{1 - x}} = \sum\limits_{k = 0}^\infty {\left( {2 + x} \right)x^k },~|x|<1$