8/17/2019 Assignment(Sequence and Series of Functions)
1/2
Sequence & Series of functions
November 8, 2015
1. Determine the limit and the nature of convergence (pointwise
or uni-form) of the sequence (f n) where f n(x) is
given by
(a) nxn, x ∈ (0, 1); (b) nnx+1
, x ∈ (0,∞); (c) sin(xn), x ∈ (0, 1)
(d) (1 + xn
)n, x ∈ R; (e) For n ≥ 2 and x ∈ [0, 1] define
f n(x) =
1− n|x − 1
n|, |x − 1
n| ≤ 1
2n0, otherwise
2. Discuss the nature of convergence (pointwise or uniform) of
the series
f n, where f n(x) is given by:
(a) 1x2+n2 , b) (−1)n (x2
+nn2 ), (c) n−2 sin(nx), (d) sin( xn2 )
3. Let f n be a sequence defined on A ⊂
R. Prove or disprove the followingstatements:
(a) If each f n is continuous, (f n)
converges uniformly on A to f , and(xn) is a
sequence in A such that (xn) converges to some x
in Athen f n(xn) → f (x).
(b) If each f n is bounded and (f n)
converges uniformly on A then (f n)is uniformly
bounded.
(c) If each f n is bounded and (f n) is
pointwise bounded then (f n) isuniformly bounded.
(d) If each f n is uniformly continuous and
(f n) converges uniformlyto f on
A then f is uniformly continuous on
A.
(e) If each f n is differentiable and
(f
n) converges uniformly on A then(f n) converges
uniformly on A.
1
8/17/2019 Assignment(Sequence and Series of Functions)
2/2
(f) If g : R → R be
uniformly continuous and if (f n) converges uni-
formly on A then (g ◦ f n) and (gf n)
converge uniformly.4. For a bounded interval [a, b] we let
C [a, b] = {f : [a, b] → R, f is
continuous}
C 1[a, b] = {f : [a, b] → R, f is
continuously differentiable}
and for f, g ∈ C [a, b] we define d(f, g)
= supx∈[a,b] |f (x) − g(x)|.
(a) Show that d is a metric and C [a, b]
is complete with respect to d.
(b) Is C 1[a, b] complete with respect to
d?
(c) For f , g ∈ C [a, b] define d1(f, g) =
supx∈[a,b]
|f (x) − g(x)|. Is C 1[a, b]
complete with respect to the metric d1?(d) Show that
C 1[a, b] is complete with respect to
d + d1(e) Show that a bounded set in (C 1[a,
b], d + d1) is compact in C [a, b].
5. The series∞
n=0
an(x − c)n is called a power series around
c, where
an, c are in R for n = 0, 1,
2 . . . . If the sequence (|an|)1
n is bounded,
set ρ = lim sup(|an|)1
n , if this sequence is unbounded set ρ =
+∞. The
radius of convergence of ∞
n=0an(x − c)
n is defined by
R :=
0 if ρ = ∞,1ρ
if 0 < ρ < ∞,∞ if ρ =
0.
The interval of convergence is the open interval (−R, R)
(a) If R is the radius of convergence of the
power series
anxn, the
show that the series
anxn converges absolutely for |x| < R and
diverges for |x| > R,
(b) Is the series converges uniformly on (−R, +R)?
(c) Is ddx
∞
n=0
anxn = ∞
n=1nanx
n−1 on (−R, R)?
(d) Determine the interval of convergence of the series
anxn, where
an is given by
i) 1nn
, ii) 2n (iii) nn
n! (iv) (−1)
n+1
n
2
