Approximation to solution of a finite sum equation In Principles of Population Genetics I encountered the equation
$$\left(1-\frac{1}{2 N}\right)^t = \sum_{i=1}^t \frac{1}{2 x_i}\tag{1}$$
, where all $N$, $t$ and all $x_i$'s are strictly positive natural numbers. From this equation, the authors say that $N$ is the harmonic mean of $[x_1, x_2, ...,x_{t-1},x_t]$. In other words, the following should be derivable from (1)
$$\frac{1}{N} = \frac{1}{t} \sum_{i=1}^t \frac{1}{x_i}\tag{2}$$
This second equation might be true only as an approximation (for large $N$ for example). I failed to prove this true.
Can you help me to prove that (2) results (directly or via an approximation) from (1)?

It's not that it follows from $(1)$, it just follows from the definition of harmonic mean which you can find here https://en.m.wikipedia.org/wiki/Harmonic_mean
In other words,
$N=\frac{1}{\frac{1}{t} \sum_{i=1}^{t} \frac{1}{x_i}}$, so $\frac{1}{N}=\frac{1}{t} \sum_{i=1}^{t} \frac{1}{x_i}$
