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?
 A: You can use the nth term test for divergence.
https://en.wikipedia.org/wiki/Term_test
If $\lim_{n \to \infty}a_n \ne 0$, then the series diverges.
A: A series $\sum_{n=1}^\infty a_n$ converges if the sequence of partial sums, i.e. $\sum_{k=1}^n a_k$, i.e.
\begin{align*}
&a_1 \\
&a_1 + a_2 \\
&a_1 + a_2 + a_3 \\
&a_1 + a_2 + a_3 + a_4 \\
&\vdots
\end{align*}
converges in the sense of sequences. In the case of the given series, the partial sums come to:
\begin{align*}
-1 &= -1 \\
1 &= -1 + 2 \\
-2 &= -1 + 2 - 3 \\
2 &= -1 + 2 - 3 + 4 \\
-3 &= -1 + 2 - 3 + 4 - 5 \\
3 &= -1 + 2 - 3 + 4 - 5 + 6 \\
&\vdots
\end{align*}
One can prove the pattern emerging with induction if they wish (or indeed by simply grouping and summing pairs of consecutive terms), but what is clear is that the sequence of partial sums is oscillating, positive and negative, while getting larger and larger. This prevents the partial sums from converging, and hence prevents the series from converging.
A: 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
