If we forget all the rules about infinte sums what am I doing wrong?

$$1-2+3-4+5-6+...=\sum\_{n=1}^{\infty} n(-1)^{n+1}$$

(with Grandi's series)

$$1,1+(-2)=-1,1+(-2)+3=2,1+(-2)+3+(-4)=-2,...$$
we got a sequence: $1,-1,2,-2,3,-3,4,-4,5,-5,...$ but that turns out to nothing and if we look to the averanges of the partial sums:

$$1,\frac{1+(-2)}{2},\frac{1+(-2)+3}{3},\frac{1+(-2)+3+(-4)}{4},\frac{1+(-2)+3+(-4)+5}{5},...$$
we got: $1,-\frac{1}{2},\frac{2}{3},-\frac{1}{2},\frac{3}{5},...$

If we take: $1+\frac{2}{3}+\frac{3}{5}+...$ we got:

$$1+\frac{2}{3}+\frac{3}{5}+...=\sum\_{n=1}^{\infty} \frac{n}{-1+2n}$$

and then we found:

$$\lim\_{n\to \infty} \frac{n}{-1+2n}=\frac{1}{2}$$

can we say that:

$$1-2+3-4+5-6+...=\sum\_{n=1}^{\infty} n(-1)^{n+1}=\frac{1}{2}$$