Let ( r=\dfrac{2n+1}{2n-1} ). Series: (r+3r^2+5r^3+\cdots)

This is an infinite-looking form in the problem statement as sum to (n) terms of an AG series in powers of (r).

[ S=\sum_{k=1}^{n}(2k-1)r^k ]

Using formula for (\sum(2k-1)x^k).

Answer: Closed form via AG sum with this (r).