if the radius of convergence of the power series ∑(n = 0 to ∞) c_nx^n is 5, what is the radius of…

if the radius of convergence of the power series ∑(n = 0 to ∞) c_nx^n is 5, what is the radius of convergence of the series ∑(n = 1 to ∞) nc_nx^(n - 1)? why? the interval of convergence remains the same when a power series is differentiated. the interval of convergence widens when a power series is differentiated. the radius of convergence decreases when a power series is differentiated. the radius of convergence remains the same when a power series is differentiated. the radius of convergence increases when a power series is differentiated. the interval of convergence narrows when a power series is differentiated. need help? read it

if the radius of convergence of the power series ∑(n = 0 to ∞) c_nx^n is 5, what is the radius of convergence of the series ∑(n = 1 to ∞) nc_nx^(n - 1)? why? the interval of convergence remains the same when a power series is differentiated. the interval of convergence widens when a power series is differentiated. the radius of convergence decreases when a power series is differentiated. the radius of convergence remains the same when a power series is differentiated. the radius of convergence increases when a power series is differentiated. the interval of convergence narrows when a power series is differentiated. need help? read it

Answer

Explanation:

Step1: Recall power - series property

The power series $\sum_{n = 0}^{\infty}c_{n}x^{n}$ and its derivative $\sum_{n = 1}^{\infty}nc_{n}x^{n - 1}$ have the same radius of convergence. This is a well - known result in the theory of power series.

Step2: Determine the radius of convergence

Given that the radius of convergence of $\sum_{n=0}^{\infty}c_{n}x^{n}$ is $R = 5$. Since the derivative of a power series has the same radius of convergence, the radius of convergence of $\sum_{n = 1}^{\infty}nc_{n}x^{n - 1}$ is also $R = 5$.

Answer:

The radius of convergence of the series $\sum_{n = 1}^{\infty}nc_{n}x^{n - 1}$ is 5. The correct option is: The radius of convergence remains the same when a power series is differentiated.