Can the radius of convergence be negative?
Definition: The radius of convergence is a non-negative number or such that the interval of convergence for the entire series $\sum_{n=0}^{\infty}a_n(x c)^n$ equals $[c R, c + R] $, $ (cR, c + R) $, $ [cR, c + R) $, $ (cR, c + R] $. …
Is the radius of convergence always positive?
In the positive case, the entire series converges absolutely and uniformly into compact sets within an open circle of radius equal to the radius of convergence, and this is the Taylor series of the analytic function to which it converges. …
Can the radius of convergence be zero?
Radius of convergence, R.
null and void. Plenty. The entire series converges for all values of x.
What does it mean if the radius of the toe is 0?
The distance between the center of the interval of convergence of a power series and its endpoints. If the series converges at a single point, the radius of convergence is 0. If the series converges at all real numbers, the radius of convergence is ∞.
What is the radius of convergence of the entire series?
If the entire series converges only at x = a, then the radius of convergence is equal to R = 0 and the interval of convergence is equal to x = a.
What is the radius of convergence of the Taylor series?
Since the Ratio Test tells us that the series converges as L 1 L 1 L 1, you are correct in your inequality statement. Since the inequality has the form ∣ x – a ∣ R | xa | if R ∣x − a∣ R, then we can say that the radius of convergence is R = 3 R = 3 R = 3.
What is the radius of convergence of the divergent series?
But the answer is that the radius of convergence will be 0 and it will converge to x = 12. …