Power Series and Convergence Tests

This video tutorial covers power series, focusing on determining the center, radius, and interval of convergence using the ratio test. It provides examples and practice problems to illustrate these concepts, helping viewers understand how to apply the ratio test to find where a power series converges. The video concludes with advanced examples and a summary of key concepts.

What is the center of a power series if it is expressed as (x + 4)^n?

• c = 4

• c = -4

• c = n

• c = 0

If the ratio test for a power series results in zero, what can be concluded about the series?

• It diverges for all x

• The radius of convergence is zero

• It converges only at x = c

• It converges for all x

In the ratio test, if the limit is infinity, what is the radius of convergence?

• Undefined

• Zero

• Infinity

• One

For the power series x^n/n!, what is the interval of convergence?

• (-1, 1)

• [0, 1]

• [1, ∞)

• (-∞, ∞)

What happens to the series n! * x^n when x = 0?

• It converges

• It is undefined

• It diverges

• It oscillates

For the series (2x - 1)^n, what is the center of the power series?

• x = 1

• x = 1/2

• x = 0

• x = -1

What is the radius of convergence for the series x^(2n)/(2n!)?

• 0

• Infinity

• 2

• 1

When checking endpoints for convergence, which test is used for alternating series?

• Integral Test

• Ratio Test

• Divergence Test

• Alternating Series Test

For the series (x - 3)^n/n, what is the interval of convergence?

• (2, 4]

• (2, 4)

• [2, 4)

• [2, 4]

In the series (-1)^(n+1) * (x - 4)^n / (n * 9^n), what is the radius of convergence?

• 4

• 13

• 3

• 9