Power Series and Functions — Question 1

*Question 1 Starting from the geometric-series formula, find a power-series representation centered at $0$ for $f(x)=1/(1-x)$. State its radius and interval of convergence, including endpoint tests.

*Question 1 – Solution

For a geometric series, $$\sum_{n=0}^{\infty}r^n=\frac1{1-r}\qquad(|r|<1).$$ Set $r=x$ to obtain $$\boxed{\frac1{1-x}=\sum_{n=0}^{\infty}x^n}.$$

The requirement $|r|<1$ becomes $|x|<1$, so the radius is $R=1$.

At $x=1$, the series is $\sum1$; at $x=-1$, it is $\sum(-1)^n$. In both cases the terms fail to approach zero.

The interval is $\boxed{(-1,1)}$. The identity holds only there, even though the rational function is defined at some points outside that interval.