Evaluate the series:

*(Putnam Competition , 2016)*

**Solution**

Since the double series the series converges absolutely we can interchange the summation. Thus:

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Evaluate the series:

*(Putnam Competition , 2016)*

**Solution**

Since the double series the series converges absolutely we can interchange the summation. Thus:

Here is a proof of why

Proof:The product telescopes, hence:

and since (that is for convergence issues) we get the result.

Yes,

another derivation is based upon the fact that every natural number has a unique binary representation.