I am working on a way of expressing some kinds of countably infinite sums in terms of easier to evaluate integrals. Many kinds of double summations are handled quite easily, however I am finding it more difficult to express using the same techniques any "single summation," such as the standard sum $\sum_{n=0}^\infty$.
It would suffice to find any kind of "nice" bijection between the naturals and some dense subset of any interval in $\mathbb{R}$. As this number will be a portion of a term in an infinite series, the "nicer" the bijection the better.
For example, Thomas Andrews (Produce an explicit bijection between rationals and naturals?) illustrated a relatively simple-to-write bijection, but it is not simplifiable in any significant way; any term in my series based on that function would of near necessity be described in terms of the prime factorization of the input, which relates to so few infinite sums as to be virtually worthless in this case. Similarly, there are many recursive results scattered throughout math.stackexchange which do not suffice.
Without the constraint that the dense subset be $\mathbb{Q}$, my hope is that this problem is tractable, and any insights against or in favor of its tractability would be appreciated.
Answer
There are other posts showing ${sin (n)} $ is dense in [-1;1]
Though I don't know if it is bijective.
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