354 CHAPTER 14. THE LEBESGUE INTEGRAL
x > 1,(√
2√
xs− x ln(
1+ s√
2x
))−(√
2s− ln(
1+ s√
2))
equals the expression
√2s(√
x−1)+ ln
1+s√
2(1+s
√2x
)x
> 0.
Therefore,(√
2√
xs− x ln(
1+ s√
2x
))>(√
2s− ln(
1+ s√
2))
and so, for s≥ 1,
exp
(−s2h
(s
√2x
))≤ exp
(−(√
2s− ln(
1+ s√
2)))
=(
1+ s√
2)
e−√
2s
Thus, there exists a dominating function for X[−√ x
2 ,∞](s)exp
(−s2h
(s√
2x
))and
limx→∞ X[−√ x
2 ,∞](s)exp
(−s2h
(s√
2x
))= exp
(−s2
)so by the dominated con-
vergence theorem,
limx→∞
∫∞
−√
x/2exp
(−s2h
(s
√2x
))ds =
∫∞
−∞
e−s2ds =
√π
See Problem 49 on Page 224 or Theorem 9.9.5. This yields a general Stirling’sformula, limx→∞
Γ(x+1)xxe−x
√2x
=√
π .
354CHAPTER 14. THE LEBESGUE INTEGRALx>I, (v2vis—xin (1 +s2)) — (v2s—In (1 +sv2)) equals the expressionV2s(/x—1) +n | v2, | 50.(oss)Therefore, (v2v —xIn (1 +s?) > (v2s—In (1 +sv2)) and so, for s > 1,(eno) sen) (asThus, there exists a dominating function for A JF | (s) exp ( sh (sy2)) and2?limy 00 A Ji] (s) exp (-» (s\/2)) = exp (—s*) so by the dominated con-vergence theorem,oO 2 ‘colim exp (-*: (+\2)) ds = / eds = VnX—}oo —4/x/2 x —ooSee Problem 49 on Page 224 or Theorem 9.9.5. This yields a general Stirling’s: Titl) _formula, lim,_,.. za = Jt.