It is a well-known fact that the harmonic series
(the sum of the reciprocals of the natural numbers)
But what about the sum of reciprocals of the prime
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.
This is best done after you have shown in class
that the harmonic series diverges.
The Math Behind the Fact:
Log(p/p-1) = Log(1 + (1/p-1)),
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
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.