HW1
1
Limits and Continuity
1) Assume that an → a and bn → b, where a and b are real numbers.
(1) Prove that an bn → ab.
[Hint]: show that |an bn − ab| ≤ |(an − a)(bn − b)| + |a(bn − b)| + |b(an − a)| using the
triangle inequality.
(2) Prove that if b 6= 0, an /bn → a/b.
[Hint] : use Theorem 1.16 to show that 1/bn → 1/b and then use the result in (1).
2) Suppose that ak → c as k → ∞ for a sequence of real numbers a1 , a2 , . . .. Prove that
n
1X
ak → c as n → ∞.
n k=1
where c is a real number.
[Hint] : Write
1
n
Pn
k=1 (ak
− c) into two sums, one for k ≤ N and one for k > N ,
1X
1X
(ak − c) +
(ak − c),
n k≤N
n k>N
and consider what happens using the definition of “ ak → c as k → ∞”.
3) Prove that if a1 , a2 , · · · is a nondecreasing (or nonincreasing) sequence, then limn an exists
and is equal to sup an (or inf n an ). We allow the possibility sup an = ∞ (or inf n an = −∞).
[Hint]: use the definition of the least upper bound and the greatest lower bound.
2
Differentiability and Taylor’s Theorem
4) Suppose f (x) is continuous in a neighborhood of x0 . Consider
lim
x→x0
f (x) − f (2x0 − x)
2(x − x0 )
(1)
(1) Is the following statement true or false? If true, prove it. If false, give a counterexample.
1
0
0
“ When f (x0 ) exists, limit in (1) also exists and it is equal to f (x0 )”.
(2) Is the following statement true or false? If true prove it. If false give a counterexample.
0
“ When limit in (1) exits, it equals f (x0 ), which also exists”.
5) Consider f (t) = log t. Taking x = a + h and a = 1 in Theorem 1.18, expand f (t) = log t
using Taylor’s theorem for all values of d ∈ {2, 3} and h ∈ {0.1, 0.01, 0.001}, and compute the
values of the remainder term rd (x, a). Describe the behavior of rd (a+h, a) as a function of h.
[Hint]: compute | rd (a + h, a) | /hd and | rd (a + h, a) | /hd+1 .
6) Definition 1.17 suggests a “first-order approximation” to a derivative of a function that
can be evaluated directly but for which no formula for the derivative is known: for a small
h,
0
f (a) ≈
f (a + h) − f (a)
.
h
(1) Construct a “second-order approximation” to a derivative by expanding both f (a+
h) and f (a − h) using Taylor’s theorem with d = 2, subtracting one expansion from
0
the other and solving for f (a) (ignore the remainder terms).
R∞
(2) Consider the gamma function Γ(x) = 0 tx−1 e−t dt where x is positive real. This
gamma function satisfies the identity Γ(x + 1) = xΓ(x). As Γ(x) grows very quickly
as x increases, it is useful to consider the log-gamma function log Γ(x) in numerical
calculations. Use the result of (1) with h = 1 to show how to obtain the following
approximation
Ψ(x) ≈ 0.5 log[x(x − 1)]
where Ψ(x) is the derivative of the log-gamma function and x > 2.
(3) Check the quality of the approximation in (2) by plotting the ratio of the approximation to Ψ(x) for all x ∈ (2, 50). Note that you can get the value of Ψ(x) if you use
digamma(x) in R.
3
Order Notation
7) For any k > −1, prove
n
X
ik ∼
i=1
2
nk+1
.
k+1
8) Prove that nβ = o ([1 + γ]n ) where β and γ are arbitrary positive constants.
9) Suppose that an ∼ bn and cn ∼ dn .
(1) Prove an cn ∼ bn dn .
(2) Show by counterexample that it is not generally true that an + cn ∼ bn + dn .
(3) Prove |an | + |cn | ∼ |bn | + |dn |.
(4) Show by counterexample that it is not generally true that f (an ) ∼ f (bn ) for a
continuous function f (x).
10) Suppose X1 , X2 , . . . , is a simple random sample from an exponential P
distribution with
density f (x) = θ exp(−θx). Consider the estimator of g(θ) = 1/θ, ĝn = ni=1 Xi /(n + 2).
Show that “ bias of ĝn ∼ c1 × variance of ĝn ∼ c2 /n as n → ∞” for some constants c1 and
c2 depending on θ.
11) The assumptions of Theorem 1.31 are as follows: an → ∞, bn → ∞, an = o(bn ) and a
convex function f (x). Find counterexamples to Theorem 1.31 by weakening the assumptions.
(1) Find an , bn and convex function f (x) with limx→∞ f (x) = ∞ such that an = o(bn )
but f (an ) 6= o[f (bn )].
(2) Find an , bn and convex function f (x) such that an → ∞, bn → ∞ and an = o(bn )
but f (an ) 6= o[f (bn )].
(3) Find an , bn and f (x) with limx→∞ f (x) = ∞ such that an → ∞, bn → ∞,
an = o(bn ) but f (an ) 6= o[f (bn )].
3