# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Productive! Topic: Algebra/NT. Level: AIME. May 23rd, 2006

Problem: (2006 Bellevue BATH Team) Evaluate .

Solution: Note that we can write the sum as

.

Using the same idea, we can write it as

.

QED.

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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

.

Furthermore, since

is an infinite geometric series of common ratio , we can say that

,

where the product is taken over all primes .