# How do you find a power series representation for  (10x)/(14+x)  and what is the radius of convergence?

Oct 8, 2015

$10 \left(\frac{x}{14} - {\left(\frac{x}{14}\right)}^{2} + {\left(\frac{x}{14}\right)}^{3} - \ldots \ldots \infty\right)$

#### Explanation:

$\frac{10 x}{14 + x} = 10 \left(\frac{x}{14 + x}\right)$

=$10 \left(1 - \frac{14}{14 + x}\right)$

=$10 - 10 \left(\frac{1}{1 + \frac{x}{14}}\right)$

Now comparing with the geometric power series $\frac{1}{1 - x} = 1 + x + {x}^{2} \ldots . \infty$ and writing $- \frac{x}{14}$ for x, the series would be,

$10 - 10 \left(1 - \left(\frac{x}{14}\right) + {\left(\frac{x}{14}\right)}^{2} - {\left(\frac{x}{14}\right)}^{3} + \ldots . . \infty\right)$

$= 10 \left(\frac{x}{14} - {\left(\frac{x}{14}\right)}^{2} + {\left(\frac{x}{14}\right)}^{3} - \ldots \ldots \infty\right)$

The implied condition of a convergent geometric series represented by $\frac{1}{1 - x}$ is -1< x <1, hence in the present case it would be -1< -x/14 <1 0r -14 < -x <14 Or 14> x >-14. Thus the radius of convergence is 14