Question

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:______

Answer 1

Answer:

Radius of the convergence is R = $\frac{1}{21}$

and,

The Interval of convergence is $-\frac{1}{21}$

Step-by-step explanation:

Given function : Σ(21x)^k

Now,

Using the ratio test, we have

R = $\lim_{n \to \infty} |\frac{21x^{k+1}}{21x^{k}}|$

or

R = $\lim_{n \to \infty} |\frac{21x^{k}\times21x^{1}}{21x^{k}}|$

or

R = $\lim_{n \to \infty} |21x^{1}|$

now,

for convergence R|x| < 1

Therefore,

$\lim_{n \to \infty} |21x^{1}|$ < 1

or

$-\frac{1}{21}$

and,

Radius of the convergence is R = $\frac{1}{21}$

and,

The Interval of convergence is $-\frac{1}{21}$