9 Power Series

Question 1

Power series

definition 9.1.1

Answer 1

A series of the form
f(x) = sum from n=0 to infinity of a_n•x^n

Where a_n In R are constants is a power series

Question 2

Theorem 9.1.2 convergence for a power series

Answer 2

For a power series f(x) = sum from n=0 to infinity of a_n • x^n
One of the following holds

• series converges only when x equals zero
• series converges for all x in the reals
• there is a constant R bigger than 0 such that the series converges absolutely if |x| is less than R and does not converge if |x| is bigger than R. ( radius of convergence)

Question 3

Definition 9.1.3. Radius of convergence

Answer 3

If the series converges only when X equals zero we say that the radius of convergence is zero

if the series converges for all x in the reals we say the radius of convergence is infinity

|x| less than R converges

|x| bigger than doesn’t

Question 4

Proposition 9.1.4 radius of convergence (finding by limits)

Answer 4

Let f(x) be a power series (defined)

Then the radius of convergence is

R =
Limit as n tends to infinity of | a_n / a_n+1|

(“Opposite of root test”)

Question 5

Example 9.1.5 find the radius of convergence of the series

f(x) = sum from n =0 to infinity of nx^n

Answer 5

Coefficient a_n =n

So limit as n tends to infinity of
| n/ n+1| = limit as n tends to infinity of n/(n+1) = limit as n tends to infinity of (1/(1+ 1/n)) =1
Thus the radius of convergence is 1.

Question 6

Lemme 9.1.6: power series and uniform convergence

Answer 6

Let
Sum from n=0 to infinity of a_n x^n

Be a power series with radius of convergence R. Let 0< S less than R.

Then the series converges uniformly on the interval
[-S,S]

Question 7

Termwise differentiation of power series theorem 9.1.7

Answer 7

Let f(x) = sum from n=0 to infinity of a_n•x^n

Be a power series with radius of convergence R then the function f is differentiable on (-R,R) with
f’(x) = sum from n=1 to infinity of n•a_n• x^{n-1}
where this series also has radius of convergence R