Convergence and Power Series Concepts

Professor Dave introduces power series, explaining their general form and how they can be represented as functions. He discusses convergence and divergence, using examples like geometric series and the ratio test. A theorem is presented to summarize convergence possibilities, including the concept of radius of convergence. An example using X to the N over N factorial demonstrates the application of these concepts, showing that this series is always convergent.

What is the general form of a power series?

• C sub N times X to the N power, from zero to infinity

• A sub N times Y to the N power, from one to infinity

• D sub N times W to the N power, from one to zero

• B sub N times Z to the N power, from zero to one

What are the constants in a power series called?

• Variables

• Coefficients

• Exponents

• Factors

How is the sum of a power series represented?

• As a single term

• As a function F of X

• As a constant value

• As a finite polynomial

What is the domain of a power series function?

• All X values for which the series is undefined

• All X values for which the series is finite

• All X values for which the series converges

• All X values for which the series diverges

When is a geometric series convergent?

• When X is between negative one and one

• When X is less than negative one

• When X is greater than one

• When X is equal to zero

What test is often used to assess convergence of a power series?

• Integral test

• Comparison test

• Ratio test

• Root test

What is the result of applying the ratio test to X minus three raised to the N power over N?

• X minus three must be greater than one

• X minus three must be less than negative one

• X minus three must be between negative one and one

• X minus three must be equal to zero

What is the radius of convergence in the third case of the theorem?

• Infinite

• Zero

• Negative

• A positive number R

What does the radius of convergence indicate?

• The interval of convergence

• The interval of divergence

• The number of terms in the series

• The sum of the series

What is the radius of convergence for a series that converges for any value of X?

• Infinite

• One

• Zero

• Negative