24 Power Series

Question 1

What type of series are Maclaurin and Taylor series?

Answer 1

They are limited series, within a specific range.

Question 2

What is a power series?

Answer 2

Infinite polynomials series,

Can be centered at 0 or any other number.

Question 3

How do you know if a series converges for all x?

Answer 3

You use the ratio test.

The limits_n→∞ of the ratio for 2 successive terms.

^{if the limit of the ratio is <1, you have absolute convergence}

^{If it is >1, you have divergence}

^{If it = 1, it is inconclusive.}

Question 4

What are the coefficients in this series?

_n=0Σ^∞ = 1 + x + ^x²/₂ + ^x³/₃ + …

Answer 4

1, 1, ½. ⅓ …

This series is centered at 0.

Question 5

What is the center of this series?

_n=0Σ^∞(-1)ⁿ(x + 1)ⁿ

Answer 5

Centered at -1 ^{(the center is at x - c)}

^{Also, this is an alternating series}

Question 6

Where is the geometric series converge?

_n=0Σ^∞xⁿ = 1 + x + x² “ …

Answer 6

It is centered at 0.

It does not converge at 1 or -1.

It does converge at 0.

It does between 1 and -1. Use the ratio test to confirm.

We already know it converges to ¹/_{(1 - x)}

Question 7

At a minimum, where do all Power Series converge?

Answer 7

At their center.

Question 8

Summarize (with examples) the 3 types of convergence.

Answer 8

Σxⁿ/n! all x, R = ∞

Σxⁿ interval (-1,1), it may or may not converge at the endpoints

Σn!xⁿ only at x = 0, R = 0( c)