Partial sum of the sequence $\frac{X-n}{n}$

Question

I have the sequence:

$S_r = \frac{x-r}{r} $

where $x$ is 2 times the greatest value for r.

How can I calculate the formula of the partial sum of this sequence ?

Answer 1

$$\sum_{r=1}^R \frac{x-r}{r} = \sum_{r=1}^R x\cdot \frac{1}{r} - 1 = x\cdot\sum_{r=1}^R r^{-1} + \sum_{r=1}^R 1 = xH_R+R$$

where $H_R$ is the $R$-th harmonic number. If $R$ is particularly large, $H_R\approx \log R + \gamma$, where $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. This brings the estimate of the partial sum for large $R$ to $x\log R + x\gamma + R$.