convergence/divergence $\sum_{n=1}^{\infty} (-1)^n\cdot\frac{6(1-n)}{n}$ 
$$\sum_{n=1}^{\infty} (-1)^n\cdot\frac{6(1-n)}{n}$$

$$lim_{n\to \infty}\frac{6(1-n)}{n}=\frac{-6n+6}{n}\approx\frac{-6n}{n}=-6$$
So $\sum_{n=1}^{\infty} (-1)^n\cdot\frac{6(1-n)}{n}$ diverge
is this valid? 

For a summation of the form $\sum_{n=x}^{\infty}a_n$ to converge in the normal sense, $\lim_{i\to\infty}a_i=0$, which is not the case.
However, as user190080 has noted, you can still evaluate the summation using varying methods.  One way to summate this is to take the averages of the partial summations, which should work because this series oscillates.
