Image by Pete Linforth from Pixabay In 1735, the renowned mathematician Leonhard Euler published the paper “De summis serierum reciprocarum” (On the sums of series of reciprocal) shown in Fig. 1 (click on this link for the full article). In this paper, the great master found the general closed formula for the sum: Euler’s astonishingly clever method “has fascinated mathematicians ever since.” Euler had previously proved the Basel problem in 1734. This result extends the Basel problem from exponent 2 to any even exponent. The three first examples are: Note that the first case in Eq. 2 is the Basel problem, also solved by Euler (in 1734), explored in one of my recent articles. But before undertaking Euler’s wonderful proof, the concept of Bernoulli numbers must be explained. This is the first part of the proof. We start by reminding the reader of the concept of Taylor series. Taylor series can be quickly defined as “a series expansion of a function about a point.” The subject of Taylor series is vast, so instead of exploring it here in detail, I will restrict myself to one case only, the expansion of the exponential function eˣ. It is given by: The radius of convergence R of eˣ is R=∞. This implies that Eq. 3 is a power series that converges for all values of x ∈ ℝ. Now, to obtain the Bernoulli numbers we perform two steps. The first is trivial: we subtract 1 from both sides of Eq. 3 and divide by x. We get:Bernoulli Numbers
valid for x ≠ 0. The Bernoulli numbers are defined by inverting this series as follows:
Equation 5: How the Bernoulli numbers are (indirectly) defined.To find the Bs, we will use a mathematical trick. Since Eq. 4 and Eq. 5 are the inverse of each other (by construction), their product is 1. We can then multiply the right-hand sides of both equations and then multiply the resulting product by n!.
After some simple algebra, we obtain a nice expression that allows us the determine the Bernoulli numbers at once:
Equation 6: Equations from which the Bernoulli number can be quickly calculated.The resulting first Bernoulli numbers are then:
Equation 7: The first Bernoulli numbers obtained from Eq. 6.The Tangent Function and Its Power Series
We now provide the second part of the proof. In this section, we will need to express the tangent of x in terms of the Bernoulli numbers. Let us first consider the following identity:
which gives:
Equation 8: Power series in terms of Bernoulli numbers.We now perform two simple steps. Substitute x on the right-hand of Eq.8 by 2ix (where i is the imaginary unit) to get:
Now, making the same substitution on the left-hand side of Eq. 8 we get:
Equation 9: The power series for x cot x written in terms of Bernoulli numbers.Then we use the trigonometric identity
and Eq. 9, to arrive at:
Equation 10: The power series for tan x written in terms of Bernoulli numbers.The Cotangent Function and Its Partial Fractions
Now to the third and last part of the proof. Using partial fractions, Euler arrived at the following expansion:
We now compare Eq. 9 and Eq. 11 replacing x by πx in the former. After some simple manipulations, we arrive at this marvelous expression:
Equation 12: The result we were after. Note that the exponent is always even. There is no equivalent expansion for odd exponents.Interestingly, the is no similar formula for odd exponents (for k=1, the sum of the reciprocals of cubes is equal to a number ~1.20 called Apéry’s constant, but there is no general formula like Eq. 12). Maybe someone reading this will find that out!
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It is a well-known fact that the harmonic series (the sum of the reciprocals of the natural numbers) diverges.
But what about the sum of reciprocals of the prime numbers?
These diverge, too!
One way to interpret this fact is that there must be a “lot” of primes—well, of course there are an infinite number of them, but not every infinite set of natural numbers has a reciprocal sum which diverges (for instance, take the powers of 2). So, while primes get sparser and sparser the farther you go out, they are not as sparse as the powers of 2.
Presentation Suggestions:
This is best done after you have shown in class that the harmonic series diverges.
The Math Behind the Fact:
Euler first noted this fact, and one proof can be obtained by taking the natural logarithm of both sides of Euler’s Product Formula, (using s=1 in that formula) and noting that the right hand side consists of terms of the form
Log(p/p-1) = Log(1 + (1/p-1)),
where Log denotes the natural log, and p is a prime. Using a Taylor series for Log, this term is itself bounded by 1/(p-1) < 1/p. Thus, if the sum of reciprocals for primes converge, then the harmonic series would converge, a contradiction.
There are many refined questions you can ask about the number of primes. See the Fun Fact How Many Primes.
How to Cite this Page:
Su, Francis E., et al. “Sum of Prime Reciprocals.” Math Fun Facts. <//www.math.hmc.edu/funfacts>.
Fun Fact suggested by:
Lesley Ward
Tags: harmonic series, medium, number theory, prime, Taylor series