When testing whether or not series converge absolutely with a convergent $p$-series, we have to test it against $\sum_{n=0}^{\infty}|a_n|$.
In one of the text book examples, I'm having trouble understanding how that absolute value works. Any insight is appreciated!
The problem is as follows:
$\sum_{n=0}^{\infty} \frac{(-1)^ni^{n+1}}{(n+1)(n+2)}$
which converges absolutely by comparison with a convergent $p$-series with $p = 2$ because, for $n \ge 1$,
$\left\lvert\frac{(-1)^ni^{n+1}}{(n+1)(n+2)}\right\rvert = \frac{1}{(n+1)(n+2)} \lt \frac {1}{n^2}$
I don't understand how the numerator is simplifying to just "1".
Note that $i$ is the imaginary unit which in complex form is $ 0+1i$
In order to find the absolute value of $ a+bi$ we find $$\sqrt {a^2+b^2} $$
Thus $$ |i|= \sqrt 1 =1 $$
As a result we have $$ |(-1)^ni^{n+1}| =1$$