Price duration under Calvo

Dear forum,
I am trying to understand the price duration calculation under Calvo pricing. In most of the resources, I have found that price duration is given as
S= (1-\theta)+2(1-\theta)\theta +3(1-\theta)\theta^2+\cdots \infty
= (1-\theta)\{ 1+ 2\theta + 3\theta^2 +\cdots \infty\}
\vdots
= \frac{1}{1-\theta}
where \theta is probability of no-price change. What I do not understand is where are the coefficients 2, 3, 4 etc. are coming from? If any one can clear this doubt, it will be really helpful.

If I remember correctly, the price S is defined as S = \sum_{n=1}^{N} n(1 - \theta)\theta^{n-1} where N is total periods and n is each period. In your case, where the total number of periods is infinity, you will then have S=1/(1-\theta).

Exactly, the numbers are the time periods for which no price adjustment has taken place.

You may find the following paper helpful:

https://www.jstor.org/stable/3838998