Problem: (2006 Bellevue BATH Team) Evaluate .
Solution: Note that we can write the sum as
Using the same idea, we can write it as
Comment: Evaluating multiple summations this way is very effective. Other examples include factoring the harmonic series
or, more generally, any integer value of the Riemann Zeta Function
is an infinite geometric series of common ratio , we can say that
where the product is taken over all primes .
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