Endless Sums and Endless Patterns, the Birth of a $1,000,000 Problem
Episode #4 of the course Great math problems for the 21st-century mind by John Robin
I will begin today with a question.
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + …
add up to? Do you believe me that the answer is infinity?
It’s not a simple answer. Mathematicians have been able to prove this only with advanced methods. This repeated sum (called a series) has a special name: the harmonic series. It has been studied to death.
Now let’s try another:
1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + 1/64 + 1/81 + 1/100 + …
Something different happens in this series. If you add the terms up, you’ll notice that eventually, it becomes 1.6449 if you round to four decimal places. Now, if you punched π2 / 6 into your calculator, you’ll notice that this is also 1.6449 if you round to four decimal places. In fact, the numbers are the same!
What exactly is happening here?
To start, notice that the above example can be written as:
1 + 1/(22) + 1/(32) + 1/(42) + 1/(52) + …
More generally, for any sum written as:
1 + 1/(2n) + 1/(3n) + 1/(4n) + 1/(5n) + …
where n is greater than 1, the series will converge to a finite number.
This series has a special name: the p-series. Just as it has been proven that the harmonic series is divergent (infinite), it has also been proven that every p-series, where n is greater than 1, converges to a finite number.
Not every number is easy to find. In the example for n = 2 above, π2) / 6 can only be calculated with some very advanced methods well beyond the scope of this course. Regardless, we can be guaranteed that there always will be a finite number that each power series will converge upon for n = 2, n = 3, n = 4, and so on.
Here are a few of them:
n = 3, i.e., 1 + 1/(23) + 1/(33) + 1/(43) + 1/(53) + …, this adds up to approximately 1.2020569 (rounded).
n = 4, i.e., 1 + 1/(24) + 1/(34) + 1/(44) + 1/(54) + …, this adds up to exactly π4/90.
n = 5, i.e., 1 + 1/(25) + 1/(35) + 1/(45) + 1/(55) + …, this adds up to approximately 1.0369278 (rounded).
n = 6, i.e., 1 + 1/(26) + 1/(36) + 1/(46) + 1/(56) + …, this adds up to exactly π6/945.
Note that by “approximately,” I mean there is no exact number we can prove the series adds up to. We can only find the number by successive calculation.
Do you have a guess what n = 7 might be? You’d be correct if you guessed it’s going to be an approximate number and pretty close to 1. In fact, it’s approximately 1.0084393. What’s happening here?
You’d be correct if you guessed that n = 8 will have π8 in it, but the pattern for the denominator is far more complicated. (There is one, though!)
It’s true that, for the p-series, there is a sort of elusive pattern—and it’s this pattern that’s at the heart of the Riemann Hypothesis. But what is the pattern exactly, and how can we put our finger on it mathematically?
To get there, I will admit that I have lied to you, in one respect. That is, I’ve assumed that n is an ordinary counting number like 1, 2, 3, 4, and so on. But mathematicians are always looking for ways to explore the boundaries beyond a given problem. And that is exactly what they did with the p-series.
Tomorrow, we will dig in more to this specific pattern behind the p-series and get one step closer to understanding the $1,000,000 problem.
Here is a link to six beautiful proofs that 1 + 1/(22) + 1/(32) + 1/(42) + 1/(52) + … = π2/6.
Introduction to Analysis of the Infinite by Leonhard Euler
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