You are somewhat close...
Our series does
not
diverge. While in principle the could be used here it turns out to be awkward, since the most obvious series to compare our series to, i.e.
¥
å
n = 2
1
3
n-1
does not work. Because of the inequality
1
3
n-1
£
1
3
n-1
-1
,
convergence of
¥
å
n = 2
1
3
n-1
does
not
imply convergence of
¥
å
n = 2
1
3
n-1
- 1
.