Progressions - Is it G.P. or H.P.?

Question

I came across the following example yesterday and could not figure it out. Any help is really appreciated.

Find the sum of the following progression from 1st to 1000th term 3, 5/3, 7/5, 9/7, 11/9, ....

Answer 1

I have only bad news

$$\frac {2k+1}{2k-1}=1+\frac {2}{2k-1}$$

So your sum depends on the harmonic sum of the odds, which depends on a finite sum of the harmonic series itself. No one has a clean closed form formula for this:

$$\sum_{n=1} ^{N} \frac 1 n $$

Much less this:

$$\sum_{n=1} ^{N} \frac{1}{2 n -1}$$