1 limit comparison with 1/3^n
2 root test
3 a_n does not go to zero
4 alternating series test
5 root test
6 limit comparison with harmonic series
7 integral test
8 ratio test
9 root test
10 root test
11 alternating series test
12 a_n does not go to zero, but this is little tricky and I talked about it in class:
you show that since pi >3, there is an integer in every interval [pi/3=2kpi,2pi/3+2kpi].
13 ratio test
14 comparison (not limit comparison) with 1/2^n
15 ratio test
16 limit comparison with 1/n
17 a_n does not go to zero
18 alternating series test
19 alternating series test
20 ratio/root test
21 root test
22 limit comparison with 1/n^2
23 limit comparison with 1/n
24  a_n does not go to zero
25 ratio test
26 ratio test
27 limit comparison with 1/n^1.5 (think of other possibilities too)
28  limit comparison with 1/n^2
29  limit comparison with 1/e^n
30 alternating series test
31 a_n does not go to zero
32 root test
33 limit comparison with 1/n^1.5
34 comparison with 1/2n
35 root test, then favorite limit to evaluate 
36 since ln n>e^2  ln n^ln n > n^2; now do comparison with 1/n^2
37 root test
38 limit comparison with 1/n