MATH 321 - EXAM #1
P ROBLEM 1. In each case below determine if the sequence or sum converges pointwise/
uniformly/neither in the given domain.
(a)
x
fn (x) = sin( ), on R.
n
(b)
∞
X
xn
, on [1, ∞).
(1 + 2x)n
n=0
P ROBLEM 2. Let f(x) = |x| and
α(x) =
0
if x < 0,
1
if x ≥ 0.
Given ε > 0, describe explicitly a partition P of [−1, 1], such that
U(P, f, α) − L(P, f, α) < ε.
P ROBLEM 3. Let f(x) be bounded and α(x) non-decreasin on [a, b]. Let P0 be one partition
of [a, b] and define
I = sup L(P∗ , f, α),
where the supremum is over all partitions P∗ that refine P0 . Prove that
Zb
I = fdα.
a
P ROBLEM 4. Let f(x) be bounded on [0, 1]. Assume that f is integrable, f ∈ R, on the
interval [0, b] for any 0 < b < 1. Prove that then f is integrable on [0, 1]. (Hint: divide [0, 1]
into two intervals [0, b] and [b, 1] for some b close to 1.)
1