Why does the series diverge? Why does the series $\sum_{n=1}^\infty(-1)^n n$ diverge?
Is it because the expanded series:
$ -1 + 2 -3 + 4 -5 + 6... $
is the difference between the infinite sum of the even and odd numbers which would be infinity?

Method 1
Try grouping 2 terms at a time
$-1+2-3+4-5+...$
$=(-1+2)+(-3+4)+(-5+6)+...$
$=1+1+1+...$
$=∞$
Method 2
$-1=-1$
$-1+2=1$
$-1+2-3=-2$
$-1+2-3+4=2$
$-1+2-3+4-5=-3$
$-1+2-3+4-5+6=3$
As we can observe the sequence doesn't seem to(and will not) converge at a particular points on the number line
