ryan dickmann math.docx

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Ryan Dickmann

Matt Baker

Math 4802

12/5/13

Problem Solving with Infinite Products

I - Why Infinite Products ?Most math inclined people are somewhat familiar with infinite series. Special series and

convergence tests are commonly taught in high school calculus, but an important related topic,

infinite products, doesnt cropup until higher level courses. Infinite products can be used,

however, to prove many interesting results similar to the unintuitive answer to the Basal

problem. In fact, Eulers well known proof that involves a comparison of a Taylorseries (an infinite sum) with an infinite product ( ). In thispaper I will explore several similar solution methods to demonstrate how they could possibly be

used on other problems.

IIInfinite Product Basics

Just like series, infinite products have several basic properties. A infinite product can

either converge or diverge. A product will converge by definition if the limit of partial products

( ) converges to a nonzero number, and otherwise it diverges. If an infiniteproduct converges, the sequence of converges to 1 (not allowing convergence to 0 is doneby convention). These results can be derived by using the convergence properties of series and


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the fact that (this is actually an useful tool itself). For two possiblyinfinite products and , ,

, and . Now with these rudimentary properties wecan investigate problems involving infinite products.

IIITelescoping Products

This technique is essentially the same as with series. The idea is to find a pattern for

canceling reciprocals such that only early and late terms remain. Then take the limit to infinity.

Here is a simple example that appears in high school math competitions.

Problem 1: Find the value of Solution:

Next is a problem from the 1977 Putnam. The problem is slightly more difficult but the process

is identical. Split the product into subproducts if needed, find canceling terms (ideally find some

functionsuch that ), and take the limits.


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Problem 2: Find the value of Solution:

The next problem is not that difficult, but it demonstrates an important technique for infinite

products. The idea is to represent a function as an expanded product so that it can be

manipulated or combined with infinite products.

Problem 3: Find a sequencesuch that Solution: Using backwards logic from the previous problems, consider

IV

Recursive Methods

The next problem solving tool involves taking a known equality and finding a way to

expand it recursively. Take for example the simple factorization of the difference of squares

which can be used to arrive at some interesting results.


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( )() ( )()()

( )()()()

( )() ( )

The following problems will be solved by setting x and y to certain values.

Problem 4: Prove that Solution: Set y = 1 in the above expression.

Rearrange the expression: ( ) ( )

Take the limit of both sides as N . The left side becomes (easily proved by L`Hopitals rule). Rearrangement proves the desired result.

Problem 5: Prove that Solution: Set and in the previously used equality.Given that and we can arrive at the result

( )


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Letting N gives us the desired result. This statement can also be proven by recursive use of the double angle formula.

Notice that by setting in the above result we can show

VThe Gamma Function

The next topic of investigation is a natural choice considering its relationship to the

factorial, a common expanded product. The following results will use the equality proved by

Gauss:

A formal proof of this will not be given, but an intuitive approach can be seen by considering

and that The equality can be rewritten with an infinite product as


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From this equality it can be easily shown that

Problem 5: Find the value of (This is called the Wallis product)Solution: Apply an index shift on the product and use the above result

Problem 1 Revisited: Find the value of Solution: Set in the above equality.

Many different ideas can be combined together to bring forth new results. Take Gauss formula

for the Gamma function and combine it with the idea of writing expanded product versions of

simple functions. First divide the numerator and denominator by n!


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Now apply the result from Problem 3: = n

VIEuler Products

Euler tackled several problems through the extremely clever use of series and infinite

products; its how he solved the Basal problem, and its also how he derived his formula for the

Riemann Zeta Function

This is essentially a statement of unique factorization, and it a great tool for problem solving.

Problem 6: Write as an infinite product.Solution:Proceeding in a similar manner as Euler, use the facts that all odd natural

numbers factor uniquely into a product of odd primes and that an odd number that is 3 modulo

4 must have all prime factors that are 3 modulo 4 occurring an odd number of times.


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Now setting

and using the sum

we arrive at the result

Note that each numerator is an odd prime and that the respective denominator is the nearest

multiple of 4.

Problem 5 Revisited: Find the value of Solution: Use Eulers infinite product for the sine function A proof of this will not be given, but an intuitive approach can be seen by factoring sine

as if it were a polynomial. Set :

Note that Now we can combine several results to get new identities. Dividing 3/4 times the result of the

Basal problem by the square of the result found from Problem 6 results in:

In each fraction the numerator and denominator differ by 1, sum to the odd primes, and the

numerator is even.


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ryan dickmann math.docx

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