Chapter 23 Power Series

Question 1

What is a Power Series? When does it converge?

Answer 1

Given a Sequence of Real Numbers, the Series Σn=0, ∞ An* x^n is a power series. It is a function of x, provided that x converges for some or all x. Of course it converges for x = 0; 0^0 = 1. For any sequence (An), one of the following hold for its power series:

a. Converges for all x ∈ ℝ. If R (radius of convergence) = +∞.
b. Series converges only for x = 0. R(radius of convergence) = 0
c. Series converges for all x in some bounded interval centered at 0; the interval may be open, half-open, or closed. R (radius of convergence) 0 < R < +∞.

Question 2

Define the main theorem of power series. Σ (An) x^n, define β and R. What is R and what is it used for?

Answer 2

β = lim sup|An|^(1/n) and R = 1 / β.
If β = 0 set R = +∞ and. 
If β = +∞ set R = 0.
Then:
i. Power series converges for |x| < R.
ii. Power Series diverges for |x| > R.

If R = 0 i. is vacuous. If R = +∞, ii is vacuous.
R is the radius of convergence for the power series.

Question 3

Simple Proof of the Power Series Theorem

Answer 3

Root Test: R = radius of convergence.
β = Limit sup|An|^(1/n)
α = lim sup|An*x^n|^(1/n) = |x| * lim sup |An|^(1/n) = β|x|
Three cases:
1. 0 < R < +∞. Then β|x| = |x|/R.
2. R = +∞. Then β = 0 and αx = 0 no matter what x (subscript) is.
3. Suppose R = 0. Then β = +∞ and αx = +∞ for x ≠ 0.

Question 4

Say β = 0, then what is R?

Answer 4

R = +∞

Question 5

Find Σ 2^(-n) * x^(3n)?

Answer 5

The power series calls the calculation of β to be involved with

Question 6

What is Lim Supremum and Lim Infimum?

Answer 6

So the limit superior is asking, how large can the tails of the sequence eventually be? Similarly, the limit inferior is asking, how small can the tails of the sequence eventually be?

Question 7

Suppose a power series Σ has finite radius of convergence R and An >= 0 for all n. Show if the series converges at R, then it converges at -R.

Answer 7

Use the Alternating Series Test. It shows us that if An converges to 0, then the alternating series converges.

Question 8

Example of Power Series with Convergence on (-1, 1]

Answer 8

An = (-1)^n / n