Someone check me?

[uncool]

Someone tell me if my math is correct?

[math] n^n[/math] acts like [math]n!e^n[/math], as follows:

[math] \frac{(n+1)^{n+1}}{n^n} = (n+1)\frac{(n+1)^n}{n^n}[/math]

[math]=(n+1)\frac{n^n+n*n^{n-1}+\frac{n(n-1)}{2}*n^{n-2}+...}{n^n}[/math]

[math]=(n+1)\frac{n^n*(1+1+\frac{1}{2}+...)+n^{n-1}*(-\frac{1}{2}-\frac{3}{6}-\frac{6}{24}-\frac{10}{120}-...)}{n^n}[/math]

The last terms can be dropped out because they contain a [math]\frac{1}{n}[/math] term.

[math]=(n+1)\frac{n^n*e}{n^n}=e*(n+1)[/math]

so after a while, the two act similarly.

In this case, similarly means that if you take the ln of both and divide them, the quotient will approach one, as shown below:

n	[math]n!*e^n[/math]   	[math]n^n[/math] 	Ratio
1	2.718281828	1            	2.718281828
2	14.7781122	4            	3.694528049
3	120.5132215	27          	4.46345265
4	1310.355601	256        	5.118576566
5	17809.57909	3125       	5.69906531
6	290468.7313	46656    	6.225752986
7	5527031.118	823543        	6.711284193
8	120192226	16777216	7.164014938
9	2940447096	387420489	7.58980792
10	79929639076	10000000000	7.992963908
11	2.38998E+12	2.85312E+11	8.376748609
12	7.79598E+13	8.9161E+12	8.743710992
13	2.75492E+15	3.02875E+14	9.095885843
14	1.04841E+17	1.1112E+16	9.434928193
15	4.27481E+18	4.37894E+17	9.762205686
16	1.85922E+20	1.84467E+19	10.07886392
17	8.59161E+21	8.2724E+20	10.38587387
18	4.2038E+23	3.93464E+22	10.684067
19	2.17115E+25	1.97842E+24	10.97416181
20	1.18036E+27	1.04858E+26	11.25678423
21	6.73795E+28	5.84259E+27	11.53248351
22	4.02944E+30	3.41428E+29	11.80174472
23	2.51923E+32	2.08805E+31	12.06499882
24	1.64351E+34	1.33374E+33	12.32263074
25	1.11688E+36	8.88178E+34	12.57498598
26	7.89361E+37	6.15612E+36	12.82237603
27	5.7934E+39	4.43426E+38	13.06508291
28	4.40947E+41	3.31455E+40	13.30336289
29	3.47599E+43	2.56769E+42	13.53744971
30	2.83462E+45	2.05891E+44	13.76755723
31	2.38864E+47	1.70692E+46	13.99388172
32	2.07776E+49	1.4615E+48	14.21660385
33	1.86382E+51	1.2911E+50	14.43589032
34	1.72257E+53	1.17566E+52	14.65189539
35	1.63885E+55	1.10251E+54	14.86476209
36	1.60375E+57	1.06387E+56	15.07462336
37	1.61299E+59	1.05551E+58	15.28160301
38	1.66614E+61	1.07591E+60	15.48581657
39	1.76632E+63	1.12595E+62	15.68737206
40	1.92054E+65	1.20893E+64	15.88637066
41	2.14044E+67	1.33088E+66	16.08290725
42	2.44369E+69	1.50131E+68	16.27707102
43	2.85634E+71	1.73438E+70	16.46894589
44	3.4163E+73	2.05077E+72	16.65861095
45	4.17892E+75	2.48064E+74	16.84614083
46	5.22536E+77	3.06803E+76	17.03160605
47	6.67588E+79	3.87792E+78	17.21507334
48	8.71052E+81	5.00702E+80	17.39660591
49	1.1602E+84	6.60097E+82	17.57626371
50	1.57688E+86	8.88178E+84	17.75410364
51	2.18607E+88	1.21921E+87	17.9301798
52	3.09002E+90	1.70677E+89	18.10454364
53	4.45176E+92	2.43568E+91	18.27724417
54	6.53462E+94	3.54212E+93	18.4483281
55	9.76961E+96	5.24745E+95	18.61784
56	1.4872E+99	7.91643E+97	18.78582241
57	2.3042E+101	1.2158E+100	18.952316
58	3.6329E+103	1.9003E+102	19.11735968
59	5.8264E+105	3.0218E+104	19.28099067
60	9.5026E+107	4.8874E+106	19.44324464
61	1.5757E+110	8.0375E+108	19.60415578
62	2.6555E+112	1.3436E+111	19.76375689
63	4.5477E+114	2.2827E+113	19.92207945
64	7.9116E+116	3.9402E+115	20.0791537
65	1.3979E+119	6.9083E+117	20.23500872
66	2.5079E+121	1.23E+120	20.38967246
67	4.5675E+123	2.2234E+122	20.54317183
68	8.4427E+125	4.0795E+124	20.69553274
69	1.5835E+128	7.596E+126	20.84678014
70	3.0131E+130	1.435E+129	20.9969381
71	5.8153E+132	2.7501E+131	21.14602983
72	1.1381E+135	5.3449E+133	21.29407772
73	2.2585E+137	1.0533E+136	21.44110339
74	4.543E+139	2.1045E+138	21.58712774
75	9.2619E+141	4.2618E+140	21.73217094
76	1.9134E+144	8.7465E+142	21.87625251
77	4.0049E+146	1.8188E+145	22.01939133
78	8.4914E+148	3.8316E+147	22.16160566
79	1.8235E+151	8.176E+149	22.30291319
80	3.9654E+153	1.7668E+152	22.44333104
81	8.731E+155	3.8662E+154	22.58287582
82	1.9461E+158	8.5652E+156	22.7215636
83	4.3908E+160	1.9208E+159	22.85940999
84	1.0026E+163	4.3597E+161	22.99643011
85	2.3165E+165	1.0014E+164	23.13263866
86	5.4154E+167	2.3274E+166	23.26804987
87	1.2807E+170	5.4724E+168	23.40267759
88	3.0635E+172	1.3016E+171	23.53653527
89	7.4114E+174	3.1312E+173	23.66963596
90	1.8132E+177	7.6177E+175	23.80199238
91	4.4851E+179	1.874E+178	23.93361686
92	1.1216E+182	4.661E+180	24.06452141
93	2.8355E+184	1.172E+183	24.19471773
94	7.2453E+186	2.9786E+185	24.32421718
95	1.871E+189	7.6514E+187	24.45303083
96	4.8825E+191	1.9863E+190	24.58116948
97	1.2874E+194	5.2102E+192	24.70864361
98	3.4295E+196	1.3809E+195	24.83546345
99	9.2291E+198	3.6973E+197	24.961639
100	2.5087E+201	1E+200   	25.08717995
101	6.8876E+203	2.7319E+202	25.2120958
102	1.9097E+206	7.5373E+204	25.33639579
103	5.3468E+208	2.1001E+207	25.46008894
104	1.5115E+211	5.9084E+209	25.58318404
105	4.3143E+213	1.6783E+212	25.7056897
106	1.2431E+216	4.8131E+214	25.82761431
107	3.6156E+218	1.3934E+217	25.94896604
108	1.0615E+221	4.0716E+219	26.0697529
109	3.145E+223	1.2008E+222	26.18998271
110	9.404E+225	3.5743E+224	26.3096631
111	2.8374E+228	1.0736E+227	26.42880153
112	8.6385E+230	3.254E+229	26.54740531
113	2.6535E+233	9.9509E+231	26.66548157
114	8.2227E+235	3.0701E+234	26.78303728
115	2.5704E+238	9.5555E+236	26.90007926
116	8.1051E+240	3E+239   	27.01661421
117	2.5777E+243	9.5005E+241	27.13264864
118	8.2683E+245	3.0344E+244	27.24818895
119	2.6746E+248	9.7744E+246	27.36324141
120	8.7243E+250	3.175E+249	27.47781213
121	2.8695E+253	1.04E+252	27.59190713
122	9.5162E+255	3.4348E+254	27.70553227
123	3.1817E+258	1.1437E+257	27.81869332
124	1.0725E+261	3.8396E+259	27.93139591
125	3.6441E+263	1.2994E+262	28.04364558
126	1.2481E+266	4.4329E+264	28.15544773
127	4.3087E+268	1.5243E+267	28.26680769
128	1.4992E+271	5.2829E+269	28.37773065
129	5.257E+273	1.8453E+272	28.48822172
130	1.8577E+276	6.4958E+274	28.59828591
131	6.6152E+278	2.3043E+277	28.70792812
132	2.3736E+281	8.2368E+279	28.81715318
133	8.5814E+283	2.9667E+282	28.92596581
134	3.1258E+286	1.0766E+285	29.03437064
135	1.1471E+289	3.936E+287	29.14237222
136	4.2405E+291	1.4497E+290	29.24997503
137	1.5792E+294	5.3792E+292	29.35718345
138	5.9239E+296	2.0105E+295	29.46400178
139	2.2383E+299	7.5693E+297	29.57043425
140	8.518E+301	2.8703E+300	29.67648501
141	3.2648E+304	1.0962E+303	29.78215814
142	1.2602E+307	4.2164E+305	29.88745764

-Uncool-