I would like to find the value of
$$\lim_{n\to \infty} \sum_{r=1}^n \frac{r}{1\cdot3\cdot5\cdot7\cdots(2r+1)}.$$
My approach is attached below.
2026-03-25 02:59:15.1774407555
Sum of a sequence.
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Now note that $\sum_{r=1}^n (T(r-1)-T(r))$ telescopes to $T(0)-T(n)$.