If the series \(\sum_{n=0}^\infty a_n\) converges conditionally, show that the radius of convergence of the power series \(\sum_{n=0}^\infty a_nx^n\) is 1.
Solution
1. \(a_n \to 0\).
\(\sum_{n\ge 0} a_n\) converges, so its terms tend to \(0\).
2. \(R \ge 1\).
for \(|x| < 1\), \(|a_n x^n| \le C|x|^n\), and \(\sum_n C|x|^n\) is a convergent geometric series; so \(\sum_n a_n x^n\) converges absolutely for every \(|x| < 1\).
3. \(R \le 1\).
if \(R > 1\), the power series converges absolutely for every \(|x| < R\), in particular at \(x = 1\), so \(\sum_n |a_n| < \infty\), contradicting conditional convergence.
4. \(R = 1\).
3.
5. Q.E.D.