Lebl-Basic-Analysis-I · Problem 2.6.8

Sequences and Series

Mathematics · power-series-radius-convergence

Problem

a) Find an example where the radii of convergence of

\sum_{n=0}^{\infty} a_n x^n

and

\sum_{n=0}^{\infty} b_n x^n

are both 1, but the radius of convergence of the sum of the two series is infinite. b) (Trickier) Find an example where the radii of convergence of

\sum_{n=0}^{\infty} a_n x^n

and

\sum_{n=0}^{\infty} b_n x^n

are both 1, but the radius of convergence of the product of the two series is infinite.