# Why is the expected time to coalesce the same as the ploidy times inbreeding effective population size?

The expected time to coalesce, in generations, is the same as the ploidy (e.g., 2 for humans) times Nef, the inbreeding effective population size, under coalescent theory. Why?

Both ploidy * Nef and the expected time to coalesce are equal to the summation from generation $$t=1$$ to infinity of (t(1-((1/(ploidy * Nef)^(t-1) * (1/(ploidy * Nef)))))), or written more prettily:

$$\sum_{t=1}^{\infty} t\left(1-\left(\frac{1}{\text{ploidy} \cdot N_{ef}}\right)^{t-1} \cdot \left(\frac{1}{\text{ploidy} \cdot N_{ef}}\right)\right).$$

Why, and what does it mean to have an "infinite" generation (in the summation)?

The time to the last coalescent (of two gene copies) is ploidy * Nef.

If you multiply this whole summation by the number of pairs of genes, the result is a new summation which represents the sum over generations of the probability that coalescence occurs in the t-th generation, or the probability of the first coalescence occuring within t generations.