## Math 508: Advanced Analysis I

Prerequisites & Review MaterialThe prerequisites for this course are Math 240 and some experience with proofs in mathematics. Note, however, that the prerequisites for Math 360 and Math 508 look very similar, Math 508 will go deeper and assume more mathematical sophistication. For instance, there will be essentially no routine homework problems whose solutions simply follow examples in the text.

You should be able to solve most of these Calculus Problems -- but some may involve real effort.

By the end of this Math 508-509 you should be able to solve all of these Analysis Problems; you already might be able to solve many of them.

Those who have not had any experience with mathematical

proofscan fill-in by reading on your own. The point is that we certainly do not emphasize proofs in Math 104-241, but they will be critical in Math 508. That is why we ask Math Majors to take the course Math 202 or Math 203 or some other course that involves proofsbeforeMath 508.

Some of the analysis material in Math 202 will be repeated in this course -- only faster and deeper.

A valuable source is Gowers: Numbers and Sets These are online problems and notes from a course at Cambridge University. While not directly related to our course, many students will find these problems and comments both enlightening and fun.Here are a few sample -- not entirely trivial -- results whose proofs you might find interesting.

- Show that the square root of 2 is not a rational number.

- There are infinitely many prime numbers. The first proof (from Euclid) is short and elementary - but very clever. You might read it somewhere, such as here: infinitely many primes.
Here is a recent variant proof by Filip Saidak: "A New Proof of Euclid's Theorem,"

Amer. Math. Monthly, Vol. 113, No. 10, Dec 2006.

Proof:Let n > 1 be a positive integer (such as n=2). Since n and n+1 are consecutive integers, they are relatively prime [that is, they do not have a common prime factor]. Hence the number N_2: = n(n + 1) must have at least twodifferentprime factors. Similarly, since the integers n(n+1) and n(n+1)+1 are consecutive, and therefore relatively prime, the number N_3: = n(n + 1)[n(n + 1) + 1] must have at least 3 different prime factors. This can be continued indefinitely, so the number of primes is infinite.

- a). Show that for any positive integer n, the number 2
^{n+2}+3^{2n+1}is divisible by 7.

b). Does this use that fact that we customarily write our integers base 10?.

c). Generalize?

- Prove that a polynomial of degree k has at most k roots. [If you prefer, assume the polynomial is real and consider only real roots].

- If a smooth function f(x) has the properties
f(0)=2, f(1)=0, and f(4)=6, show that there is a point c with 0 < c < 4 where f"(c) > 0.

Better yet, find some explicit number m > 0 so that for this c we have f"(c) > m. From the graph, this is certainly obvious intuitively -- but what about a proof?

- Prove that the function sin x is not a polynomial. That is, there is no polynomial
p(x) = a with real coefficients so that sin x = p(x) for all real numbers x. In your proof you may use any standard properties of the function sin x._{0}+ a_{1}x + ... + a_{n}x^{n}

- Find positive integers k and N so that:
1 + 2 +... + k = (k+1) + ... + N Are there infinitely many k and N?

What about the similar problem1 + 2 +... + (k - 1) = (k+1) + ... + N ?

- Assume the earth is a sphere and you know the longitude and latitude of two cities, both at sea level. Find a formula for the (flying) distance between them. [Since the radius of the earth is about 4000 miles and the height of Mt. Everest is roughly 5.5 miles, there is essentially no error if you assume all cities are at sea level.]

You might also find it useful (but not essential) to read a bit in the Math 202 text:

Mathematical Thinking, Problem-Solving and Proofs

by John D'Angelo and Douglas West,

second edition, Prentice Hall.

[This book is in Penn's Math-Physics Library, Call Number: QA39.2 .D25 2000]

Stuff to read: Part I , Chapters 1 and 2, Skip parts II and III. Continue with Part IV, chapters 13, 14, 15.