UCLA Calculus Worksheet

Problem 1. Suppose f : R ! R is a continuous function which can be uniformly approximated by

polynomials on R. Show that f is itself a polynomial.

(Hint: If Pn and Pm are polynomials, then so is Pn ?? Pm. Assuming jPn(x) ?? Pm(x)j < ” for all x 2 R,

what does that tell you about Pn ?? Pm? Sub-hint: how do polynomials behave at innity?)

Remark. Note that this exercise implies that the Weierstrass approximation theorem fails in general when

we are not dealing with a compact domain.

Problem 2. Consider the space C(R;R) of continuous real-valued functions.

(a) Given any f 2 C(R;R), show that there exists a sequence of polynomials (Pn)n that converges

pointwise to f, i.e. limn!1 Pn(x) = f(x) for every x 2 R.

(b) Show that one can moreover choose the sequence (Pn)n in such a way that the convergence is uniform

on any compact subset of R, i.e. for every K R compact, Pn ! f uniformly on K. (Note: the

sequence Pn does not depend on K!)

Problem 3. Exercise 4.1.2 from the textbook. (Note that we saw most examples you will need in class.

Explain brie

y for every example why it satises the requirements.)

Problem 4. Show that the function f : R n f1g ! R given by f(x) = 1

1??x is real analytic on its domain.

Given a 6= 1, what is the radius of convergence of the power series of 1

1??x centered at a?

Problem 5. Discuss the convergence of the following series by calculating the radius of convergence R

and checking what happens at the boundary points” x = R.

1X

n=1

(??1)n+1

n

xn and

1X

n=0

1

3nn2 xn:

Do these series converge uniformly on any of the intervals [??r;R]; [??R; r], or [??R;R] where 0 < r < R?

Problem 6. Let a > 0. As another application of the contraction mapping theorem, we will nd an

exponentially fast algorithm to calculate

p

a: dene

f(x) =

1

2

x +

a

x

:

(a) Show that

p

a is a xed point of f.

(b) Show that f([

p

a;1)) [

p

a;1).

(c) Show that f is a contraction on [

p

a;1). Conclude that starting with any x0 2 [

p

a;1), the sequence

(xn)n dened by xn = f(xn??1) converges to

p

a.

(Hint: you can use the following fact, which follows easily from the fundamental theorem of calculus.

For f dierentiable:

jf(x) ?? f(y)j

max

xy

f0()

jx ?? yj :

Optional: Prove this!)

(d) Take a = 2, x0 = 2 and compute xn for n = 1; 2; 3.