Watching the wheels

 

      N                    F(N)=N^N       G(N)=F(N)/F(N-1)   G(N)-G(N-1) 
     1
1
     
 
2
4
4
3
27
6.75
2.75
4
256
9.481481
2.731481
5
3125
12.20703
2.72555
6
46656
14.92992
2.722889
7
823543
17.65138
2.721465
8
16777216
20.372
2.720613
9
387420489
23.09206
2.720063
10
10000000000
25.81175
2.719687
11
2.85312E+11
28.53117
2.719419
12
8.9161E+12
31.25039
2.719221
13
3.02875E+14
33.96946
2.719071
14
1.1112E+16
36.68841
2.718954
15
4.37894E+17
39.40727
2.718861
16
1.84467E+19
42.12606
2.718786
17
8.2724E+20
44.84478
2.718725
18
3.93464E+22
47.56346
2.718674
19
1.97842E+24
50.28209
2.718632
20
1.04858E+26
53.00069
2.718596
21
5.84259E+27
55.71925
2.718565
22
3.41428E+29
58.43779
2.718539
23
2.08805E+31
61.15631
2.718516
24
1.33374E+33
63.8748
2.718496
25
8.88178E+34
66.59328
2.718479