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

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

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

Which of following intervals of convergence cannot exist?

Possible Answers:

For any  such that , the interval 

For any , the interval  for some 

Correct answer:

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 #2 : Radius And Interval Of Convergence Of Power Series

Find the interval of convergence of  for the series .

Possible Answers:

Correct answer:

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 #3 : Radius And Interval Of Convergence Of Power Series

Find the interval of convergence for  of the Taylor Series .

Possible Answers:

Correct answer:

Explanation:

Using the root test

 

and

. T

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

This occurs when x=5.

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