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

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