Concept preview

What you will be able to do

1. 01Apply the ratio test to a power series $\sum c_n(x-a)^n$ to find its radius of convergence $R$, the half-width of the interval on which the series converges.
2. 02Explain why the ratio test is inconclusive exactly at the endpoints $x=a\pm R$, and test each endpoint separately with an appropriate convergence test to obtain the exact interval of convergence.
3. 03Recognise the three possible outcomes for a power series — $R=0$ (a single point), a finite $R$, or $R=\infty$ (all real $x$) — and state the interval as open, closed, or half-open based on the endpoint results.

Start with the object itself. A power series centred at $a$ is $\sum_{n=0}^{\infty} c_n(x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots$, where the numbers $c_n$ are the coefficients and $a$ is the centre. Think of $x$ as a dial. When you set the dial to $x=a$ every term after the first vanishes and the series trivially converges to $c_0$, so $x=a$ always works. As you turn the dial away from $a$, the powers $(x-a)^n$ grow, and at some point they grow so fast that the coefficients can no longer tame them and the series diverges. The distance you can move away from $a$ before that happens is the radius of convergence $R$.