Is there a mistake in this page on asymptotic expansions?

Question

I think there is an error in section 4.3 of this page - http://aofa.cs.princeton.edu/40asymptotic/

The author says that by taking $x = -\frac{1}{N}$ in the geometric series

$\frac{1}{1-x} = 1 + x + x^2 + ...$ as $x \to 0$

gives

$\frac{1}{N + 1} = \frac{1}{N} - \frac{1}{N^2} + ...$ as $N \to \infty$

However if the equation I get if I make that substition is

$\frac{1}{1 + \frac{1}{N}} = 1 - \frac{1}{N} + \frac{1}{N^2} - ...$ as $N \to \infty$

So has the author made a mistake or I have I misunderstood something?

Answer 1

$$\frac{1}{1 + \frac{1}{N}}=\frac{N}{N+1}\ne\frac{1}{N+1}\qquad(N\ne1)$$

Answer 2

The author didn't make a mistake, you just hadn't notice he "skipped" a step

$\frac{1}{1+\frac{1}{N} } = \frac{N}{N+1} = 1-\frac{1}{N+1} $

Combined with the results you provided you get the desired result.