# AP Calculus BC : Radius and Interval of Convergence of Power Series

## Example Questions

### Example Question #21 : Series In Calculus

Which of following intervals of convergence cannot exist?

For any , the interval  for some

For any  such that , the interval

Explanation:

cannot be an interval of convergence because a theorem states that a radius has to be either nonzero and finite, or infinite (which would imply that it has interval of convergence ). Thus,  can never be an interval of convergence.

### Example Question #1 : Radius And Interval Of Convergence Of Power Series

Find the interval of convergence of  for the series .

Explanation:

Using the root test,

Because 0 is always less than 1, the root test shows that the series converges for any value of x.

Therefore, the interval of convergence is:

### Example Question #1 : Taylor Series

Find the interval of convergence for  of the Taylor Series .

Explanation:

Using the root test

and

. T

herefore, the series only converges when it is equal to zero.

This occurs when x=5.