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# Tag Archives: Sums

## Binomial coefficients as multiple sum

Prove that

**Solution**

The binomial coefficient in the RHS enumerates the subsets of size of . The LHS does the same thing, but choosing first the largest element of , then its second-to-largest element , until choosing its smallest element .

## On the factorial

Let denote the Möbius function and denote the floor function. Prove that:

**Solution**

The RHS equals

since for .

## Sum over all positive rationals

For a rational number that equals in lowest terms , let . Prove that:

**Solution**

First of all we note that

Moreover for we have that

Hence for we have that

## On the dubious function

The *dubious function * is defined as follows : and

Evaluate the sum

## Double summation

Evaluate

**Solution**

*Coming soon!*