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and
N+1
e −x − e −(N+1)x The Coefficients of q(n) 29
−sx
e ds.
x 1
Hence, by Fubini, we may interchange and obtain, for our expression,
the elementary sum
N 1 N+1
t 1 ds
− dt +
k + t − 1 2 s
k 1 0 1
N
k 1
(k − 1) log − 1 + log(N + 1)
k − 1 2
k 1
N

(k − 1) log k − (k − 1) log(k − 1) − N
k 1
1
+ log(N + 1)
2
N log N − log N − log(N − 1) − ··· − log 1 − N

1
+ log(N + 1)
2
1
N log N − log N! − N + log(N + 1).
2
√ N
What luck! This is equal to log N+1(N/e) and so, by Stirling’s
N!
1
formula, indeed approaches log √ .
2π
(Stirling’s formula was used twice and hence needn’t have been
√
used at all! Thus we ended up not needing the fact that C 2π
√
n
in the formula n! ∼ C n(n/e) since the C cancels against a C in
√
the denominator. The n! formula with C instead of 2π is a much
simpler result.)