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 ,