Consider this series using the root test

consider the series:
∞
∑ an
n=1
where an= (-1)^n((ln(n))/n)^n
determine whether this series converges or diverges

Your use of parenthesis is quite perplexing. I'm going to assume that you meant the following:

oo
∑ (-1)^n * (ln(n)/n)^n
n=1

Applying the Root Test, yields:

lim n-->oo | (-1)^n * (ln(n)/n)^n |^(1/n),
lim n-->oo | -1 * (ln(n)/n) |,
lim n-->oo | -ln(n)/n | (direct sub. here, gives indeterminate form oo/oo).

Applying L'Hopital's rule, yields:

lim n-->oo | -(1/n) | = 0.

Since 0 < 1, the series converges.

Hope this helped.