notation for Sumation of Sumation for only for odd iterations

Question

I need to write a summation in summation whether the inner summation should iterate from one through all odd numbers to the teration of the outer summation which goes from 1 to $\infty$... Something like...

$$ \sum_{i = 1}^{\infty} \sum_{1 \le k \le i, k\text{ odd}}^{i} \text{expression} $$

This notation I found in: Sigma notation only for odd iterations

However, I am still wondering, if there is any other, more elegant way how to do it...

Answer 1

I'd argue that using $2k-1$ is the elegant way to do it: $$\large\sum_{i=1}^\infty\sum_{k=1}^{\lceil i/2\rceil}(\text{expression using $2k-1$})$$ What you wrote in your post is one alternative, though it can be written a bit more cleanly as: $$\large\sum_{i=1}^\infty\sum_{\substack{k=1\\k\text{ odd}}}^i(\text{expression using $k$})$$

Answer 2

You could use $$\sum\limits_{k=1}^{\lfloor\frac{i+1}{2}\rfloor}$$ then change the $k$ inside your summation to $2k-1$. Not sure if that is the kind of thing you are looking for.