contestada

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence.

Σ(21x)^k

The radius of convergence is R =______

a. The interval of convergence is {x: x=______ }
b. The Interval of convergence is:______

Respuesta :

Answer:

Radius of the convergence is R = [tex]\frac{1}{21}[/tex]

and,

The Interval of convergence is [tex]-\frac{1}{21}<x<\frac{1}{21}[/tex]

Step-by-step explanation:

Given function : Σ(21x)^k

Now,

Using the ratio test, we have

R = [tex]\lim_{n \to \infty} |\frac{21x^{k+1}}{21x^{k}}|[/tex]

or

R = [tex]\lim_{n \to \infty} |\frac{21x^{k}\times21x^{1}}{21x^{k}}|[/tex]

or

R = [tex]\lim_{n \to \infty} |21x^{1}|[/tex]

now,

for convergence R|x| < 1

Therefore,

[tex]\lim_{n \to \infty} |21x^{1}|[/tex] < 1

or

[tex]-\frac{1}{21}<x<\frac{1}{21}[/tex]

and,

Radius of the convergence is R = [tex]\frac{1}{21}[/tex]

and,

The Interval of convergence is [tex]-\frac{1}{21}<x<\frac{1}{21}[/tex]

Otras preguntas