This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: } P  < 1 sup { s n  : n > N } P < 1 sup { s n  : n > N } < P + 1 . Hence, we can bound { s n  : n N } by M = max { s 1  ,  s 2  ,...,  s N  ,P + 1 } . 12.14. (a) By Theorem 12.2 we have lim sup  n !  1 /n lim inf  n !  1 /n lim inf  ( n + 1)! n !  = lim inf  n +1  = so by the lim inf/lim sup theorem, we have that ( n !) 1 /n diverges to . 1 2 (b) Let s n = 1 n n ( n !). Then,  s n +1 s n  = n n ( n +1) n . for all n . We claim that lim n n n ( n +1) n = 1 e by the quotient theorem for limits because lim n n ( n + 1) n = lim 1 ( n +1) n n n = lim 1 (1 + 1 n ) n = 1 e . By Corollary 12.3, we then have lim 1 n ( n !) 1 /n = lim  1 n n ( n !)  1 /n = lim  s n +1 s n  = e1 ....
View Full
Document
 Spring '09

Click to edit the document details