Irrational Exuberance

Suppose n is a natural number and r a rational number, 0 < r < 1. Consider the series

Is it irrational?

Need a hint? Click here.

Click here for the complete solution.

Hint 1

In general it is almost impossible to tell from an infinite series whether the sum is rational or irrational. Instead, try to find a closed-form expression for this series.

Hint 2

Part of the general term can be factored out to get the binomial coefficient {1/n \choose k}.

Hint 3

We can rearrange the general term to get

Therefore by the binomial theorem the sum is

The Rest of the Solution

Now let's investigate when x might be rational. Let's write r=b/c for positive integers b and c. Then the expression for x rearranges to

If x is rational, then the first term in the equation is rational, and an integer, so it has to be a perfect nth power of an integer, say

and so

which is just Fermat's Last Theorem! Conversely if there are such integers a,b,c then we can find a rational x. So the problem of finding r such that x is rational is the same as Fermat's Last Theorem. So there are two cases:

• n > 2: the only solution is r=0.
• n = 2: the solutions are r=b/c where b and c are a side and the hypotenuse of an integral right triangle.

Irrational Series

Consider these very similar looking series; are their values rational or irrational?

Series (1) is rational, because it is a telescoping series in disguise,

Series (3) and series (5) are irrational (in fact transcendental), because they are equal to \pi^2/6 and \pi^4/90 respectively (a famous result of Euler's). Series (4) is irrational, as was proved by Roger Apéry in 1979. Series (2) and series (6) are still unknown.

It's very difficult to determine whether a given series sums to a rational or irrational value. One general result is that if the series converges "rapidly" then it converges to an irrational value. For example, we can prove that e = \sum_{n=0}^\infty 1/n!, the base of natural logarithms, is irrational because its infinite series converges rapidly. Another example, discovered by Liouville in 1851, is that the series

where k is an integer greater than 1, converges to a transcendental value (these were the first known examples of transcendental numbers).

Series (3) through (6) are particular values of the Riemann zeta function \zeta(s)=\sum_{n=1}^\infty 1/n^s a very important function in Prime Number Theory. Euler discovered an explicit formula for its values at positive even integer arguments, and we can tell from his formula that the values are transcendental. Euler and many others attempted to get an explicit formula for the zeta function at positive odd integer arguments, but no one has succeeded. Apéry's proof that series (4) is irrational works directly from the series, without evaluating it. In 2000 T. Rivoal announced a proof that the zeta function is irrational for infinitely many positive odd integer arguments.

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, Oxford University Press, 2008. Proves the irrationality of e as Theorem 47 on pp. 53-54 (read it on Google Books here) and the transcendence of Liouville's series is Theorem 192 on pp. 207-208.
• T. W. Körner, Exercises for Fourier Analysis, Cambridge University Press, 1993. Sketches a proof of Apéry's theorem on pp. 172-175. You can read it online at Google Books here.
• Wikipedia has a proof of the irrationality of e.
• Click here to view the original problem submitted by Pingo.