# How do you find a power series representation for (1+x)/(1-x)  and what is the radius of convergence?

Oct 1, 2015

Simplify $\frac{1 + x}{1 - x}$ to express in terms of the power series for $\frac{1}{1 - x}$ and find:

$\frac{1 + x}{1 - x} = 1 + {\sum}_{n = 1}^{\infty} 2 {x}^{n}$

with radius of convergence $1$

#### Explanation:

$\frac{1 + x}{1 - x} = \frac{2 - \left(1 - x\right)}{1 - x} = \frac{2}{1 - x} - 1$

$\frac{1}{1 - x} = {\sum}_{n = 0}^{\infty} {x}^{n}$ with radius of convergence $1$

To see that for yourself, notice that:

$\left(1 - x\right) {\sum}_{n = 0}^{\infty} {x}^{n}$

$= {\sum}_{n = 0}^{\infty} {x}^{n} - x {\sum}_{n = 0}^{\infty} {x}^{n}$

$= {\sum}_{n = 0}^{\infty} {x}_{n} - {\sum}_{n = 1}^{\infty} {x}^{n}$

$= 1$

.. if the sums converge.

Now the sum ${\sum}_{n = 0}^{\infty} {x}^{n}$ is a geometric series with common ratio $x$. So it converges when $\left\mid x \right\mid < 1$.

So:

$\frac{1 + x}{1 - x} = 1 + {\sum}_{n = 1}^{\infty} 2 {x}^{n}$

with radius of convergence $1$