Cal1-TD5(2017-18))

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Institute of Technology of Cambodia Calculus I (2017-18)
I1–TD5
Sequence of Functions
1. Over the given interval I determine the sets of pointwise and uniform convergence, and
the limit function, for:
(a) fn(x) =
nx
1 + n3x3
, I = [0,+∞)
(b) fn(x) =
x
1 + n2x2
, I = [0,+∞)
(c) fn(x) = e
nx, I = R
(d) fn(x) = nxe
−nx, I = R
(e) fn(x) =
1
(1 + x)1+1/n
, I = R+
(f) fn(x) =
4nx
3nx + 5nx
, I = R
(g) fn(x) =
sin(nx+ 3)√
n+ 1
, I = R
(h) fn(x) = n
2xn, I = (0, 1)
(i) fn(x) = x
2 exp
(
− sin x
n
)
, I = R
(j) fn(x) =
sin(nx)
n
√
x
I = R∗+
2. Let (fn) be a sequence of function defined by
fn(x) =

sinx
x(1 + nx)
, if x 6= 0
1, if x = 0
(a) Study the pointwise and uniform convergent of (fn) on [0, π].
(b) Let a ∈]0, π[. Study the pointwise and uniform convergent of (fn) on [a, π].
3. Study pointwise and uniform convergence for the sequence of function:
fn(x) = nx(1− x2)n, x ∈ [−1, 1]
Does the following formula hold?
lim
n→∞
∫ 1
0
fn(x)dx =
∫ 1
0
lim
n→∞
fn(x)dx.
4. Study pointwise and uniform convergence for the sequence of function:
fn(x) = arctannx, x ∈ R
Tell whether the formula
lim
n→∞
∫ 1
a
fn(x)dx =
∫ 1
a
lim
n→∞
fn(x)dx.
holds, for a = 0 or a = 1/2.
5. Suppose 0 < a < b. Prove that
lim
n→∞
∫ b
a
(
1 +
x
n
)n
e−xdx = b− a.
Dr. Lin Mongkolsery 1/5
Institute of Technology of Cambodia Calculus I (2017-18)
6. Study pointwise and uniform convergence for the sequence of function:
fn(x) = cos
(
xe−nx
2
)
, x ∈ R
then deduce limit of the sequence
In =
∫ 1
0
fn(x) dx
7. Find an example of a sequence (fn) of continuous functions from [0, 1] to R which
converges pointwise to 0 but for which limn→∞
∫ 1
0
fn(x)dx 6= 0.
8. Let (fn) be a sequence of continue function on R. Show that, if the sequence (fn)
uniformly convergent on R to function f , then the sequence of function (fn ◦ fn)
converge pointwie on R.
9. Let g : R −→ R. Suppose that g is differentiable, g(0) = 0, g′(0) 6= 0 and limx→±∞ g(x) =
0. For each n ∈ N define fn : R −→ R by
fn(x) =
g(nx)
n
Prove that fn −→ 0 uniformly, but f ′n 9 0.
10. Show that the sequence of functions (x+1/n) converge uniformly on R, but ((x+1/n)2)
does not converge uniformly.
11. Let A ⊂ R, and let (fn) and (gn) be uniformly convergent sequences of functions from
A to R. Prove that (fn + gn) converges uniformly.
12. Suppose that fn : [a, b] → R converges uniformly to f where f is continuous on [a, b].
Suppose further that a sequence (xn), whose elements in [a, b], coverges to x. Show
that
fn(xn)→ f(x).
13. Let fn : R→ R defined by
fn(x) =
√
x2 + 1/n
Show that fn is in class C1 and (fn) converges uniformly on R to a function f which is
not in class C1.
14. Let f : R→ R be two times differentiable and f ′′ is bounded. Show that the sequence
of function
gn(x) = n (f(x+ 1/n)− f(x))
converges uniformly to f ′.
15. Let (fn) be a sequence of function defined on R+ by
f0(x) = x and fn+1(x) =
x
2 + fn(x)
for all n ∈ N
Study pointwise and uniform convergence of (fn) on R+.
Dr. Lin Mongkolsery 2/5
Institute of Technology of Cambodia Calculus I (2017-18)
Series of Functions
16. Determine the set of pointwise convergence of the series:
(a)
∞∑
n=1
(n+ 2)x
n3 +
√
n
(b)
∞∑
n=1
(
1 +
x
n
)n2
(c)
∞∑
n=1
1
xn + x−n
, x > 0
(d)
∞∑
n=1
xn
xn + 2n
, x 6= −2
(e)
∞∑
n=1
(−1)nnx sin 1
n
(f)
∞∑
n=1
(
n−
√
n2 − 1
)x
17. Determine the set of pointwise and uniform convergence of the series
∞∑
n=2
enx. Compute
its sum, where defined.
18. Setting fn(x) = cos
x
n
, check
∑∞
n=1 f
′
n(x) converges uniformly on [−1, 1], while
∑∞
n=1 fn(x)
converges nowhere.
19. Determine the set of pointwise and uniform convergence of the series:
(a)
∞∑
n=1
n1/x (b)
∞∑
n=1
(lnn)x
n
(c)
∞∑
n=1
[
(n2 + x2)1/(n
2+x2) − 1
]
20. Show that the following series converge uniformly on the indicated set I.
(a)
∞∑
n=1
(−1)n+1x
n
n
, I = [0, 1]
(b)
∞∑
n=1
sin(nx)
n
, I = [α, 2π−α], 0 <α< π
(c)
∞∑
n=1
sin(n2x) sin(nx)
n+ x2
, I = R
(d)
∞∑
n=1
(−1)n+1 e
−nx
√
n+ x2
, I = [0,∞)
(e)
∞∑
n=1
(−1)n+1arctan(nx)
n+ x2
, I = R
(f)
∞∑
n=1
(−1)n+1 cos(x/n)√
n+ + cosx
, I = [−R,R].
21. Suppose that
∑∞
n=0 anx
n converges for x = −4 and diverges for x = 6. What can be
said about the convergence or divergence of the series?
(a)
∞∑
n=0
an (b)
∞∑
n=0
an7
n (c)
∞∑
n=0
an(−3)n (d)
∞∑
n=0
(−1)nan9n
22. Let p be a positive integer. Determine, as p varies, the radius of convergence of
∞∑
n=0
(n!)p
(pn)!
xn
23. Find radius and set of convergence of the power series:
Dr. Lin Mongkolsery 3/5
Institute of Technology of Cambodia Calculus I (2017-18)
(a)
∞∑
n=1
xn√
n
(b)
∞∑
n=0
(−1)nxn
n+ 1
(c)
∞∑
n=0
nx2n
(d)
∞∑
n=2
(−1)n x
n
3n lnn
(e)
∞∑
n=0
n2(x− 4)n
(f)
∞∑
n=0
n3(x− 1)n
10n
(g)
∞∑
n=1
(−1)n (x+ 3)
n
n3n
(h)
∞∑
n=1
n!(2x− 1)n
(i)
∞∑
n=1
nxn
1.3.5 . . . (2n− 1)
(j)
∞∑
n=1
(−1)nx2n−1
2n− 1
24. The function
J1(x) =
∞∑
n=1
(−1)nx2n+1
n!(n+ 1)!22n+1
is called Bessel function of order 1. Determine its domain.
25. Determine the convergence set of the series:
(a)
∞∑
n=1
(
2
3
)n
(x2 − 1)n (b)
∞∑
n=1
1
n
√
n
(
1 + x
1− x
)n
(c)
∞∑
n=1
n+ 1
n2 + 1
2−nx
2
26. Expand in Maclaurin series the following functions, computing the radius of conver-
gence of the series thus obtained:
(a) f(x) =
x3
x+ 2
(b) f(x) = ln
1 + x
1− x
(c) f(x) = 2x
(d) f(x) =
√
2− x
(e) f(x) =
1
2 + x− x2
(f) f(x) = ex sinx
27. Expand the functions below in Taylor series around the point x0, and tell what is the
radius of the series:
(a) f(x) =
1
x
, x0 = 1 (b) f(x) =
√
x, x0 = 4 (c) f(x) = lnx, x0 = 2
28. Find the radius of convergence R of the power series
∞∑
n=0
x2n+1
(2n+ 1)!!
and show that its sum f satisfies the equation f ′(x) = 1 + xf(x), x ∈ (−R,R).
29. Determine the radius of convergent R and calculate for x ∈ (−R,R), the sum
(a) f(x) =
∞∑
n=0
xn
(2n)!
(b) g(x) =
∞∑
n=0
sinhn
(2n)!
x2n
Dr. Lin Mongkolsery 4/5
Institute of Technology of Cambodia Calculus I (2017-18)
(c) h(x) =
∞∑
n=0
(n2 + n+ 1)xn (d) k(x) =
∞∑
n=0
n2 + n+ 1
n!
xn
30. (a) Determine the radius of convergent R of the series
∞∑
n=1
x3n
(3n)!
(b) For x ∈ (−R,R), find the closed form of the series.
31. Let f be a function defined on domain D by
f(x) =
∞∑
n=1
x2n+2
n(n+ 1)(2n+ 1)
(a) Find the radius of convergent R and domain of convergent D of f .
(b) Determine the closed form of f .
(c) Deduce the value of the series
∞∑
n=1
1
n(n+ 1)(2n+ 1)
32. Let f be a function defined on domain D by
f(x) =
∞∑
n=1
(−1)n−1x2n
n(2n− 1)
(a) Find the radius of convergent R and domain of convergent D of f .
(b) Determine the closed form of f .
(c) Deduce the value of the series
∞∑
n=1
(−1)n−1
n(2n− 1)
33. For arctanx is analytic on (−1, 1), establish the following equalities:
arcsinx = x+
∞∑
n=1
(2n− 1)!!
(2n)!!(2n+ 1)
x2n+1, arctanx =
∞∑
n=0
(−1)n
2n+ 1
x2n+1.
Deduce that
π
6
=
1
2
+
∞∑
n=1
(2n− 1)!!
22n+1(2n)!!(2n+ 1)
and
π
4
=
∞∑
n=0
(−1)n
2n+ 1
.
34. (a) Find the radius of convergent of the series
∞∑
n=1
cos(nα)
n
xn, and
∞∑
n=1
sin(nα)
n
xn
(b) For |x| < 1, calculate the sum C(x) =
∞∑
n=1
cos(nα)
n
xn.
(c) Show that for |x| < 1, we have
∞∑
n=1
sin(nα)
n
xn = arctan
(
x sinα
1− x cosα
)
Dr. Lin Mongkolsery 5/5