You cannot prove this because it isnottrue: The radius of convergence of the series might belargerthan , but it cannot be strictlysmaller.

For a trivial example of this just take and for all . The two series individually have a radius of convergence equal to 1 but the power series with coefficients converges for all whatever.

The radius of convergence cannot be strictly smaller, because if , then the two series both converge absolutely and therefore their sum also converges absoluely.