2006 SMT/Team Problems/Problem 15
Solution
Denote as . To begin, let = .
Let's notice that:
And that:
Notice that all the fractions, except and will cancel out with their reciprocals from the next term(i.e cancels with , with ).
Therfore:
So we have now proven that . We will save this for later. For the second part, we will use the famous identity (discovered by Euler) that: .
Plugging in , we see that: .
Therefore, we can split into two infinite products, one of prime numbers, and one of composite numbers
Noticing that:
We can conclude that:
Therefore, the answer is ,