Chapter 7 - Definitions

Question 1

What is the definition for radius of convergence?

Answer 1

The radius of convergence of a power series (the sum from n=0 to infinity asubscriptnx^n) is R:= sup{|x|: |the sum from n=0 to infinity asubscriptnx^n|converges}.

If the set is unbounded above, we say that R = infinity

Question 2

What does the radius of convergence tell us?

Answer 2

It tells us almost everything about the subset of the real numbers on which it converges.

Question 3

What are the sum series for exp, sin and cos?

Answer 3

exp: real numbers to the real numbers is
exp(x)=lim from n=0 to infinity of asubscriptn/n!

sin: real numbers to the real numbers is
sin(x)=lim from n=0 to infinity of (-1)^n*x^(2n+1)/(2n+1)!

cos: real numbers to the real numbers is
cos(x)=lim from n=0 to infinity of n*x^(2n)/(2n)!

Question 4

What is the definition for termwise derivative?

Answer 4

Given a power series f(x)=lim from n=0 to infinity of asubscriptnx^n, its termwise derivative is the power series
fwith a top hat=lim from n=1 to infinity of nasubscriptn*x^(n-1)
= lim from n=0 (n+1)asubscrpt(n+1)x^n
- derivative is still just a name here -

Question 5

What is the definition/equivalent for a to the power x?

Answer 5

Let a>0 and x exist in the real numbers. Then, a^x = exp (xlna)

Question 6

What is the definition for analytic?

Answer 6

Let U be a subset of the real numbers and be open, Then f: U to the real numbers is analytics if, for each xsubsrcipt0 exist in U, there exists epsilon>0 and a power series lim from n=0 to infinity of asubscriptn(x-xsubscript0)^n which converges to f(x) for all x that exists in (xsubscript0 - epsilon, xsubscript0 + epsilon)

Question 7

What is the definition for polynomial function?

Answer 7

Given a polynomial p(x) = asubscript0 + asubscript1x + …+ asubscriptk*x^k, we define the function fsubscriptp: real numbers to the real numbers such that
fsubscriptp(x) = 0, x<or=0
= p(1/x)exp(-1/x), x>0 the function is smooth but not analytic - (except p(x)=0)