1.6. Exercises 41
Hint: Write
log n +
1
2
log
1
2
=
n+ 1
2
1
2
1
x
dx,
and estimate
1
j

j+ 1
2
j− 1
2
dx
x
.
Exercise 1.3. Show that there is a c 0 such that the following is
true. For every real number r and every integer n,
(1.32)
e−cr2/n

er
1
r
n
n

ecr2/n.
Exercise 1.4. Find constants a1,a2 such that the following is true
as n ∞,
1
1
n
n
=
e−1
1 +
a1
n
+
a2
n2
+ O
(
n−3
)
.
Exercise 1.5. Let Sn be a one-dimensional simple random walk and
let
pn = P{S2n = 0 | S0 = 0}.
Show that
(1.33) pn+1 = pn
2n + 1
2n + 2
,
and hence
pn =
1 · 3 · 5 · · · (2n 1)
2 · 4 · 6 · · · (2n)
.
Use the relation (1.33) to give another proof that there is a
c such that, as n ∞,
pn
c

n
.
(Our work in this chapter shows, in fact, that c = 1/

π, but you do
not need to prove this here.)
Exercise 1.6.
Show that if X is a nonnegative random variable, then
lim
n→∞
E[X 1{X n}] = lim
n→∞
E[X n] = E[X].
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