Approximate average customer lifetime using churn Average customer lifetime can be approximated using formula: $1/Churn$ (churn is a real number in interval $(0;1)$ The same can be achieved using more complex formula: $\sum_{n=0}^\infty (1-Churn)^{(n-1)}$ Wolphram alpha calculates for $Churn = 0.0205$ sum of $48.7805$. This matches $1/0.0205 = 48.7805$ Does it mean that $\sum_{n=0}^\infty (1-Churn)^{(n-1)} = 1/Churn$ ?

Yes, you have a geometric distribution of how long customers stay. Your sum is a geometric series Using the sum of a geometric series you can prove the equality you want.