3) Functions defined by power series

Question 1

What is the interval of convergence

Answer 1

The set of all x ∈ R where a power series converges

Question 2

What is the RoC when nth term is multiplied by n

Answer 2

Question 3

Describe the proof for the RoC when nth term is multiplied by n

Answer 3

Question 4

What is the domain of a power series

Answer 4

The interval of convergence of the power series

Question 5

What is Lipschitz continuity (L-continuity/conditon)

Answer 5

Question 6

What is the the L-condition for the function x^n

Answer 6

Question 7

Describe the proof of the L-condition for the function x^n (Not needed)

Answer 7

Question 8

Is a function determined by a power series continuous

Answer 8

A function determined by a power series is continuous on the open interval of its RoC (-R,R)

Question 9

What is the exponential function

Answer 9

Question 10

What are some properties of the exponential function

Answer 10

• exp is continuous on R (Power Series Continuous Therem)
• exp(x+y) = exp(x) exp(y) (Convolution Theorem for Series)
• exp(x) > 1 for positive x and exp(x) > 0 for all x ∈ R
  (exp(x)exp(−x) = exp(0) = 1 and so exp(-x) = 1/ exp(x))
• exp is strictly increasing (let x < y, Since y − x > 0, exp(y − x) > 1 and so exp(y) > exp(x))
• The image of exp is (0, ∞) ( Inverse Function Theorem)