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

Radius of the convergence is R =
`(1)/(21)`

and,

The Interval of convergence is
`-(1)/(21)<x<(1)/(21)`

Explanation:

Given function : Σ(21x)^k

Now,

Using the ratio test, we have

R =
`\lim_(n \to \infty) |(21x^(k+1))/(21x^(k))|`

or

R =
`\lim_(n \to \infty) |(21x^(k)*21x^(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

`-(1)/(21)<x<(1)/(21)`

and,

Radius of the convergence is R =
`(1)/(21)`

and,

The Interval of convergence is
`-(1)/(21)<x<(1)/(21)`