Math 3210-1
HW 18
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Pointwise and Uniform Convergence
n
1. Let fn (x) = xn for x ∈ [−1, 1]. Find f (x) = lim fn (x) and determine whether or not the convergence
is uniform on [−1, 1]. Justify your answer.
2. Let fn (x) =
x
x+n
for x ≥ 0.
(a) Show that f (x) = lim fn (x) = 0 for all x ≥ 0.
(b) Show that if t > 0, the convergence is uniform on [0, t].
(c) Show that the convergence is not uniform on [0, ∞).
3. If (fn ) and (gn ) converge uniformly on a set S, prove that (fn + gn ) converges uniformly on S.
4. Determine whether or not the given series of functions converges uniformly on the indicated set. Justify
your answers.
X
√
(a)
n−x for x > 2
(b)
X x2
for x ≥ 5
n2
Applications of Uniform Convergence
nx
for x ∈ [0, 1]. Show that the sequence (fn ) does not converge uniformly on [0, 1]
1 + nx
by using Theorem 117.
5. Let fn (x) =
6. Let fn (x) =
n + sin nx
for x ∈ R.
3n + sin2 nx
(a) Show that (fn ) converges uniformly on R.
Z π
(b) Use Theorem 118 to evaluate lim
fn (x) dx.
n→∞
0
7. Using Corollary 9, integrate the geometric series
1
= 1 + t + · · · + tn + · · ·
1−t
term by term from −x to x, where x ∈ (−1, 1), and obtain a series for log
Uniform Convergence of Power Series
8. (a) Find the function given by the series
∞
X
n=1
n2 xn for |x| < 1.
∞
∞
X
X
n2
n2
(b) Evaluate
and
.
n
2
2n
n=1
n=2
9. (a) Show that
∞
X
1
=
(−1)n x2n for |x| < 1.
1 + x2
n=0
1+x
.
1−x
(b) Show that arctan x =
∞
X
(−1)n x2n+1
for |x| < 1.
2n + 1
n=0
(c) Show that the series for arctan x in part(b) also holds when x = 1.
(d) Use part (c) to find a series whose sum is π.
Z x
∞
X
(−1)n−1 x2n
for |x| < 1.
10. (a) Show that
arctan t dt =
2n(2n − 1)
0
n=1
(b) Show that the formula in part (a) also holds for x = 1.
(c) Assuming that the series 1 −
value.
1
2
−
1
3
+
1
4
+
1
5
−
1
6
−
1
7
+ · · · is convergent, use part (b) to find its