Does Analysis Appear on Putnam?

Question

I was recently searching past Putnam problems, I believe I saw some sort of Real Analysis, but I am not sure.

Is Real Analysis (Mathematical Analysis) a topic, which appears on the Putnam Exam? Which things in general? Thanks!

Answer 1

In principle, you should never need the fancier tools of real analysis. A solid background in calculus up to multivariable is enough. You should, however, be aware of some definitions and basic theorems, like the epsilon-delta definitions of limit/continuity/derivative, some theorems about convergence (definition of Cauchy sequence, various series tests, Bolzano-Weierstrass, etc.), some of which may or may not be covered in a calculus course depending on the university and the type of calculus sequence. I believe complex analysis is not usually required, though you will need to be comfortable with the idea of functions of a complex variable, polynomials in particular.

In practice, having experience with analysis will make you much more comfortable with certain problems. There is usually at least one problem that requires you to come up with some inequality, either as the end goal or as an intermediate result in some other problem. If nothing else, brush up on the major inequalities: AM-GM-HM, Schwarz, Holder, Jensen's, Cauchy's inequality (maybe with $\epsilon$), etc. etc.

Many of the problems that are analytically flavored will have multiple solutions, some of which can pull in some more advanced analysis. Thus it may be helpful to know the statements of the Lebesgue dominated convergence theorem or the monotone convergence theorem for integrals, and some related results (e.g. differentiation under the integral sign). However, these theorems should never be required to come up with a solution; problems that can be solved using these will always have more elementary solutions. You probably won't need anything more advanced than this.

In summary, if you only know calculus you're still in good shape with some practice. But knowing more won't hurt you, practically speaking. (Psychologically, knowing more can restrict how you view a problem, but that's a different story.)