Homework # 2, Math 413
Due Friday, January 24, 2014
This homework set has ve (5) problems. Some of them are routine while others require
more thought. You are encouraged to work with others and to ask
Homework # 6 Solutions, Math 413
1. If we toss a fair coin 2n times, there are a total of 22n possible outcomes
(two possibilites on each toss). The total number of outcomes with exactly
n heads is
(2
Homework # 3 Solutions, Math 413
1. Find the pointwise limit for each of the sequences of functions dened
below, then decide if the convergence is uniform.
(a) fj (x) = jx(1 x)j
x [0, 1]
It is clear o
Homework # 4 Solutions, Math 413
1. The hypotheses on this one are exactly those of Dinis theorem, which you
proved on HW #3. Thus, we know that the sequence fj actually converges
uniformly to f on [a
Homework # 5 Solutions, Math 413
1. Here we examine the function
2
f (x) =
e1/x
0
if x = 0
if x = 0
It is clear that we may dierentiate f as many times as we like at any
x = 0. To prove that f is inni
Homework # 2 Solutions, Math 413
1. By the single partition characterization of Riemann integrability, we know
that given > 0, there does exist a partition P such that
k
sup f inf f
Ij
Ij
j=1
j < .
Cl
Homework # 1 Solutions, Math 413
1) Suppose f is Riemann integrable on [a, b], say with
b
A=
f (x) dx
a
Then by denition there exists > 0 such that for any two partitions P, P
with mesh m(P ), m(P ) <
Homework # 5, Math 413
Due Friday, February 21, 2014
This homework set has seven (7) problems. Most of them are routine while some require
more thought. You are encouraged to work with others and to a
Homework # 7, Math 413
Due Friday, March 14, 2014
This homework set has eight (8) problems. Most of them are routine while some require
more thought. You are encouraged to work with others and to ask
Homework # 7 Solutions, Math 413
1. Bolzano-Weierstrass Theorem in Rn : Suppose |xj | M for all j N.
Then cfw_xj has a convergent subsequence.
Proof: We proceed by induction on n. Let S = cfw_n N : B