Power Series and Functions — Question 1

Question 1 – Solution

Step 1: Use the geometric prototype.

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

Step 2: Carry over the convergence condition.

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

Step 3: Test the endpoints.

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

Conclusion.

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