I was fascinated by studying infinite series in calculus; the idea of adding up infinitely many elements and possibly having a finite number was not intuitive for me at first. These problems can help in bridging the gap between philosophical and practical aspects of mathematics. For instance, the famous Zeno’s paradox argues the impossibility of movement based on the infinite divisibility of space. With this paradox and assuming you have a certain distance to travel, you must first travel half that distance and then you must travel half the remaining distance, and then again half of the remaining distance and so on forever. So looking at this problem it seems as if we might not ever be able to get anywhere. However, if you were to stand a certain distance away from a wall and then attempt to do this, at some point you would undoubtedly walk yourself right into the wall. So is mathematics lying to us?
The answer no, this problem can be represented as an infinite series, 1/(2n) from n = 1 to n = infinity, which represents adding up the following elements (1/2 + 1/4 + 1/8 + 1/16 …). The problem is that we could never physically add up the infinitely many terms on a piece of paper, or even in all the books filling a library. What we can do is add up a portion of the sum, called a partial sum, to some nth element. Once we find the partial sum we can use calculus to take the limit as n grows increasing larger to infinity and see if we arrive, or converge, to a finite number. So for the series above, the partial sum can be represented as 1 – 1/(2n), and after taking the limit as n approaches infinity the second term goes to 0 and we are left with 1, a finite number. Even though you are adding up infinitely many elements, after traveling deep into your series, the elements that you are appending to the sum become so small that they are negligible to the total sum. These types of problems appear often in mathematics and especially integral calculus. In order to find the area under a curve in calculus we end up taking Riemann sums, drawing rectangles under the curve to approximate the area. Then we examine these sums when we take more and more smaller rectangles (infinitely many) that better approximate the area and eventually our our approximation turns into the actual solution. Infinite series appear all over in mathematics.
Perhaps the most famous mathematical problem is the unsolved Riemann Hypothesis. This problem deals with finding the roots of the Riemann Zeta Function which itself is an infinite series. This function maps a complex number s to the series (1/1s + 1/2s + … + 1/ns) from n = 1 to n = infinity, or more formally written as:
This function is defined for all complex numbers where the real part is greater than 1. This means that plugging in a complex number with real part greater than 1, the series will add up to, or converge to, a finite number. The mathematician for which the Zeta function was named after, Bernhard Riemann, was able to prove an analytic continuation of this function so that it is defined for all complex numbers s, except s = 1.
An analytic continuation is a technique used in complex analysis to extend the domain of an analytic function. The method is to find a new function which is defined over a larger domain. If the new function equals the old function on the intersection between the original function’s domain and the new function’s domain, then the new function is called an analytic continuation of the original function. So in the case of the Riemann Zeta function, an analytic continuation would be a new function which is defined exactly the same as the Zeta Function for complex values where the real part is greater than 1. However, the analytic continuation would also be defined on a larger domain, where complex values with real part less than one also have a value. For instance, plugging in the value s = -1 into the original Riemann Zeta Function would result in the series 1 + 2 + 3 + 4 + … off to infinity, which most of us will clearly recognize as being a divergent series. However, it is often said, especially in physics and string theory, that 1 + 2 + 3 + 4 + … = -1/12. This is because by use of Riemann’s analytic continuation, ζ(-1) = -1/12. Using this technique, all complex numbers besides s = 1, are defined for the Riemann Zeta Function. For s = 1 the function outputs the harmonic series, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + … This series is not always clear to those who are first learning about convergence and divergence. The function 1/n tends to zero as it grows, but using methods in calculus we are able to prove that it is actually a divergent series and that it sums to infinity. Riemann was not able to account for this series in his analytic continuation, so the value s = 1 is a single pole, or singularity, where the function is undefined.
Some of the roots of the Zeta function are obvious to mathematicians, such as all of the negative even integers. Riemann conjectured that all other non-trivial roots would have a real part of 1/2, meaning on the complex plane all nontrivial roots would fall on the line Re[1/2]. However, while we have found many roots on this critical line (millions) this conjecture has yet to be proven. This is one of the Millennium Problems, so whoever does end up proving or disproving this will receive a prize of one million dollars awarded from the Clay Mathematics Institute. This problem is not simply about finding where a function equals, but has deep connections with number theory and specifically with the distribution of prime numbers where other proofs are yet to be complete without it.
Convergent infinite series are really interesting and not necessarily intuitive upon a first learning about them. Sometimes you can worry yourself thinking about the notion of infinity or the infinitesimal, it is hard to relate to something so large or small. The idea of zero is a little easier because we are more comfortable with the idea of nothing, but “a number smaller than any other number” is hard to wrap your mind around. So adding up these numbers that are practically nothing, but still something, seems like such an incredible mathematical achievement to me.
Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.
Davis, Philip J., and Reuben Hersh. The Mathematical Experience. Birkhauser Boston, 1981. Print
Sondow, Jonathan and Weisstein, Eric W. “Riemann Zeta Function.” http://mathworld.wolfram.com/RiemannZetaFunction.html