1 +
l~v+°
So, for j large, ej is about —l/2j. This is unfortunate since
oo
1
From this one can show that the limit lim defined above does
NOT exist.
Let us try to improve the approximation, by including a power
of n, i.e., we will approximate n\ by nne~nns, where s is some num-
ber independent of n. What value of s shall we choose? Let yn =
n![n
n
e~
n
n
s
]
- 1
, CQ = 1 and cn = yn-i/yn,
s o
that
- i
2/n+i =°i-"
c
n+i
We have already seen that
3=2
3 + 1
bi
= l - ~ + 0
2j \ r
For a fixed s, we can do another Taylor expansion and see that
j + i
We now choose s = 1/2 so that the (1/j) terms will cancel. If s = 1/2,
But, J^ j
- 2
oo, so we can conclude that the limit
n n r
lim
T/"1
= lim TT Cj =
e"1
lim TT 1 + O
exists. We finally have shown the following: There exists a positive
number L such that
lim '-—= = L.
n-*oc
nne~ny/n
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