11. Are also analytic, since their defining power series have the same radius of convergence as the original series. 12. Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. 13. The radius of convergence is infinite if the series converges for all complex numbers " z ". 14. These formulas are similar to the Cauchy-Hadamard theorem for the radius of convergence of a power series. 15. It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series. 16. The distance from the center to either of those points is 2?, so the radius of convergence is 2?. 17. Since the radius of convergence of this power series is infinite, this definition is applicable to all complex numbers z. 18. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere. 19. The radius of convergence is 1 / " e ", as may be seen by the ratio test. 20. Be a power series with real coefficients " a " " k " with radius of convergence 1.
