Solution to Problem A4 from 1999 Putnam Math Contest

By. Ajay Mohan

Abstract

The Putnam is a math contest set out for undergraduate math students from all backgrounds, organised the the Mathematical Association of America(MAA). It consists of two sections, A and B, with 6 problems a section. Problems increase in difficulty, where problem A6 and B6 are the hardest. Moreover, the Putnam is a great way for students to express their mathematical knowledge in a creative and exiting way! In this article I present a solution to problem A4 from the 1999 Putnam.

Problem

Sum the series

Solution

Let

We can rewrite this as

This is equivalent to writing out 2 copies of the double sum itself, hence we have

Notice that for the RHS expression we have interchanged the variables(Fairly standard trick). Now we want to be able to combine both of these expressions into a single expression. To do so we must make the denominators the same. Hence, we multiply the denominator and numerator of the LHS expression by , similarly for the RHS we multiply by , therefore:

Simplifying and factoring we get that

Now notice that both expressions in the numerator and denominator are equal, thus we can cancel them out leaving

We can now extract the sums to get a single sum each

The first few terms of the series are

This is not a geometric series, so we need to rely upon a little formatting adjustments. We can write the series out in rows like so

These are all geometric so it is easy to evaluate each one line by line. The first two lines for example would be

Therefore what we essentially have is

This then evaluates to

Taking a look at our sum

Notice how the series we just evaluated is the same as the LHS? This means that we can now write out our final answer to be

Commentary on solution given

The solution provided at https://kskedlaya.org/putnam-archive/1999s.pdf uses a shortcut. To explain this shortcut let us make this series a bit more interesting, we have that

This is a power series representation of the series of interest. The first few terms are

Notice that this is now a geometric series so we can apply the formula:

By differentiating and using the chain rule we have that

By subbing in 1 we get

It then follows the desired answer as:

End Remarks

Evaluating series of this kind is often difficult as one has to spot patterns or make the problem easier by reducing it to something familiar. In this case, we reduced our series in such a way that we could apply the geometric formula. Of course, there are many other techniques from analysis we could have twisted and used to evaluate this series.