Let the power series series \(\sum_{n=0}^\infty a_nx^n\) and \(\sum_{n=0}^\infty b_nx^n\) have radii of convergence \(R_1\) and \(R_2\), respectively.
Solution
1. Cauchy–Hadamard formula: the radius of convergence of \(\sum a_n x^n\) is \(R = 1/\limsup_{n \to \infty} |a_n|^{1/n}\) (with \(1/0 = \infty\), \(1/\infty = 0\)).
By the root test, since \(\limsup |a_n x^n|^{1/n} = |x| \limsup |a_n|^{1/n}\), the series converges absolutely when \(|x| < R\) and diverges when \(|x| > R\), because then its terms do not tend to \(0\).
2. If \(R_1 \neq R_2\), the radius of convergence of \(\sum (a_n + b_n) x^n\) is \(\min\{R_1, R_2\}\), and in general it is at least \(\min\{R_1, R_2\}\). See P-E4WZN.
3. The radius of convergence of \(\sum a_n b_n x^n\) is at least \(R_1 R_2\). See P-FYGQ6.