Originally Posted by

**xifentoozlerix** the problem is to find the power series about the origin of the given function:

$\displaystyle \frac{{1 + z}}

{{1 - z}},{\text{ }}\left| z \right| < 1$

this is my work...

$\displaystyle \frac{{1 + z}}

{{1 - z}} = \left( {1 + z} \right)\left( {\frac{1}

{{1 - z}}} \right) = \left( {1 + z} \right)\left( {\sum\limits_{n = 0}^\infty {z^n } } \right) $$\displaystyle = \sum\limits_{n = 0}^\infty {z^n } + \sum\limits_{n = 0}^\infty {z^{n + 1} } = \sum\limits_{n = 0}^\infty {\left( {z^n + z^{n + 1} } \right)} = 1 + 2\sum\limits_{n = 0}^\infty {z^{n + 1} }$

my question is if this is the form required or is there is some way to get the 1 back inside the sum? or would it be better to leave it as $\displaystyle

\sum\limits_{n = 0}^\infty {\left( {z^n + z^{n + 1} } \right)} $?