Bui Xuan DieuMath4 Exercises 1.1 Double Integrals 1.1.1 Double Integrals in Cartesian coordinate Exercise 1.1.. 1.1.4 Double Integrals in polar coordinate Exercise 1.12... 1.3 Triple Int
Trang 1Hanoi University of Science and Technology Dr Bui Xuan Dieu
Math4 Exercises
1.1 Double Integrals
1.1.1 Double Integrals in Cartesian coordinate
Exercise 1.1 Evaluate
a) RR
D
x sin(x + y)dxdy, where D =(x, y) ∈ R2: 0 ≤ y ≤ π2, 0 ≤ x ≤ π2
b) RR
D
x2(y − x) dxdy where D is the region bounded by y = x2 and x = y2
c) RR
D
|x + y|dxdy, D :=(x, y) ∈ R2||x ≤ 1| , |y| ≤ 1
d) RR
D
p|y − x2|dxdy, D :=(x, y) ∈ R2||x| ≤ 1, 0 ≤ y ≤ 1
e) RR
[0,1]×[0,1]
ydxdy (1+x 2 +y 2 )3
f) RR
D
x2
y 2dxdy, where D is bounded by the lines x = 2, y = x and the hyperbola xy = 1
1.1.2 Change the order of integration
Exercise 1.2 Change the order of integration
a)
1
R
−1
dx
1−x2
R
−√1−x 2
f (x, y) dy
b)
1
R
0
dy
1+√
1−y 2
R
2−y
f (x, y) dx
c)
2
R
0
dx
√ 2x
R
√ 2x−x 2
f (x, y) dx
d)
√ 2
R
0
dy
y
R
0
f (x, y) dx+
2
R
√ 2
dy
√
4−y 2
R
0
f (x, y) dx
1.1.3 Change of variables
Exercise 1.3 Evaluate I =RR
D
4x2− 2y2 dxdy, where D :
1 ≤ xy ≤ 4
x ≤ y ≤ 4x
Exercise 1.4 Evaluate
I =
Z Z
D
x2sin xy
y dxdy, where D is bounded by parabolas
x2= ay, x2= by, y2= px, y2= qx, (0 < a < b, 0 < p < q)
Trang 2Exercise 1.5 Evaluate I =
D
xydxdy, where D is bounded by the curves
y = ax3, y = bx3, y2= px, y2= qx, (0 < b < a, 0 < p < q)
Hint: Change of variables u = x3
y , v = yx2 Exercise 1.6 Prove that
1
Z
0
dx
1−x
Z
0
ex+yy dy = e − 1
2 . Hint: Change of variables u = x + y, v = y
Exercise 1.7 Find the area of the domain bounded by xy = 4, xy = 8, xy3= 5, xy3= 15
Hint: Change of variables u = xy, v = xy3, (S = 2 ln 3)
Exercise 1.8 Find the area of the domain bounded by y2= x, y2= 8x, x2= y, x2= 8y
Hint: Change of variables u = yx2, v = xy2, (S = 279π2 )
Exercise 1.9 Hint: Change of variables y = x3, y = 4x3, x = y3, x = 4y3
Exercise 1.10 Prove that
Z Z
x+y≤1,x≥0,y≥0
cos x − y
x + y
dxdy = sin 1
2 . Hint: Change of variables u = x − y, v = x + y
Exercise 1.11 Evaluate
I =
Z Z
D
r x
a+
r y b
dxdy,
where D is bounded by the axes and the parabolapx
a +py
b = 1
1.1.4 Double Integrals in polar coordinate
Exercise 1.12 Express the double integral I =RR
D
f (x, y) dxdy in terms of polar coordinates, where D
is given by x2+ y2≥ 4x, x2+ y2≤ 8x, y ≥ x, y ≤√3x
Exercise 1.13 EvaluateRR
D
xy2dxdym where D is bounded by
x2+ (y − 1)2= 1
x2+ y2− 4y = 0
Exercise 1.14 Evaluate
a) RR
D
D
|x − y|dxdy,
where D : x2+ y2≤ 1
Exercise 1.15 EvaluateRR
D
dxdy (x 2 +y 2 ) 2, where D :
4y ≤ x2+ y2≤ 8y
x ≤ y ≤ x√
3
Exercise 1.16 EvaluateRR
D
xy
x 2 +y 2dxdy, where D :
x2+ y2≤ 12, x2+ y2≥ 2x
x2+ y2≥ 2√3y, x ≥ 0, y ≥ 0
Trang 31.2 Applications of Double Integrals
Exercise 1.17 Compute the area of the domain D bounded by
a)
y = 2x, y = 2−x,
y = 4
b)
y2= x, y2= 2x
x2= y, x2= 2y
c)
y = 0, y2= 4ax
x + y = 3a, (a > 0)
d)
x2+ y2= 2x, x2+ y2= 4x
x = y, y = 0
e) r = 1, r = √2
3cos ϕ
f) x2+ y22
= 2a2xy (a > 0)
g) x3+ y3= axy (a > 0) (Descartes leaf) h) r = a (1 + cos ϕ) (a > 0) (Cardioids)
Exercise 1.18 Compute the volume of the object given by
a)
3x + y ≥ 1, y ≥ 0
3x + 2y ≤ 2,
0 ≤ z ≤ 1 − x − y
b) V :
0 ≤ z ≤ 1 − x2− y2
y ≥ x, y ≤√
3x
c) V :
x2+ y2+ z2≤ 4a2
x2+ y2− 2ay ≤ 0
Exercise 1.19 Compute the volume of the object bounded by the surfaces
a)
z = 4 − x2− y2
2z = 2 + x2+ y2
b)
z = x
2
a2 +y
2
b2, z = 0
x2
a2 +y
2
b2 =2x a
c)
az = x2+ y2
z =px2+ y2
Exercise 1.20 Find the area of the part of the paraboloid x = y2+ z2 that satisfies x ≤ 1
1.3 Triple Integrals
1.3.1 Triple Integrals in Cartesian coordinate
Exercise 1.21 EvaluateRRR
V
x2+ y2 dxdydz, where V is bounded by the sphere x2+ y2+ z2= 1 and the cone x2+ y2− z2= 0
1.3.2 Change of variables
Exercise 1.22 Evaluate
a) RRR
V
(x + y + z)dxdydz, where V is bounded by
x + y + z = ±3
x + 2y − z = ±1
x + 4y + z = ±2
b) RRR
V
(3x2+ 2y + z)dxdydz, where V : |x − y| ≤ 1, |y − z| ≤ 1, |z + x| ≤ 1
c) RRR
V
dxdydz, where V : |x − y| + |x + 3y| + |x + y + z| ≤ 1
Trang 41.3.3 Triple Integrals in Cylindrical Coordinates
Exercise 1.23 EvaluateRRR
V
x2+ y2 dxdydz, where V :
x2+ y2≤ 1
1 ≤ z ≤ 2 Exercise 1.24 EvaluateRRR
V
zpx2+ y2dxdydz, where:
a) V is bounded by: x2+ y2= 2x and z = 0, z = a (a > 0)
b) V is a half of the sphere x2+ y2+ z2≤ a2, z ≥ 0 (a > 0)
Exercise 1.25 Evaluate I =RRR
V
p
x2+ y2dxdydz where V is bounded by:
x2+ y2= z2
z = 1
Exercise 1.26 EvaluateRRR
V
dxdydz
√
x 2 +y 2 +(z−2) 2, where V :
x2+ y2≤ 1
|z| ≤ 1
1.3.4 Triple Integrals in Spherical Coordinates
Exercise 1.27 EvaluateRRR
V
x2+ y2+ z2 dxdydz, where V :
1 ≤ x2+ y2+ z2≤ 4
x2+ y2≤ z2 Exercise 1.28 EvaluateRRR
V
p
x2+ y2+ z2dxdydz, where V : x2+ y2+ z2≤ z
Exercise 1.29 Evaluate RRR
V
zpx2+ y2dxdydz, where V is a half of the ellipsoid x2a+y2 2 +zb22 ≤ 1, z ≥
0, (a, b > 0)
Exercise 1.30 EvaluateRRR
V
x2
a 2 +yb22 +z2
c 2
dxdydz , where V : x2
a 2 +yb22 +z2
c 2 ≤ 1, (a, b, c > 0)
Exercise 1.31 EvaluateRRR
V
p
z − x2− y2− z2dxdydz, where V : x2+ y2+ z2≤ z
Exercise 1.32 EvaluateRRR
V
(4z − x2− y2− z2)dxdydz, where V is the sphere x2+ y2+ z2≤ 4z
Exercise 1.33 EvaluateRRR
V
xzdxdydz, where V is the domain x2+ y2+ z2− 2x − 2y − 2z ≤ −2 Exercise 1.34 Evaluate
I =
Z Z Z
V
dxdydz (1 + x + y + z)3, where V is bounded by x = 0, y = 0, z = 0 và x + y + z = 1
Exercise 1.35 Evaluate
Z Z Z
V
zdxdydz,
where V is a half of the ellipsoid
x2
a2 +y
2
b2 +z
2
a2 ≤ 1, (z ≥ 0)
Exercise 1.36 Evaluate
a) I1=RRR
B
x 2
a 2 +yb22 +zc22
, where B is the ellipsoid xa22 +yb22 +zc22 ≤ 1
Trang 5b) I2 =
C
zdxdydz, where C is the domain bounded by the cone z2 = Rh2(x2+ y2) and the plane
z = h
c) I3=RRR
D
z2dxdydz, where D is bounded by the sphere x2+y2+z2≤ R2and the sphere x2+y2+z2≤ 2Rz
d) I4 =RRR
V
(x + y + z)2dxdydz, where V is bounded by the paraboloid x2+ y2 ≤ 2az and the sphere
x2+ y2+ z2≤ 3a2
Exercise 1.37 Find the volume of the object bounded by the planes 0xy, x = 0, x = a, y = 0, y = b, and the paraboloid elliptic
z = x
2
2p +
y2
2y, (p > 0, q > 0).
Exercise 1.38 Evaluate
I =
Z Z Z
V
p
x2+ y2+ z2dxdydz,
where V is the domain bounged by x2+ y2+ z2= z
Exercise 1.39 Evaluate
I =
Z Z Z
V
zdxdydz,
where V is the domain bounded by the surfaces z = x2+ y2 and x2+ y2+ z2= 6
Exercise 1.40 Evaluate
I =
Z Z Z
V
xyz
x2+ y2dxdydz, where V is the domain bounded by the surface (x2+ y2+ z2)2= a2xy and the plane z = 0
2.1 Definite Integrals depending on a parameter
Exercise 2.1 Compute
a) lim
y→0
1+y
R
y
dx
y→0
2
R
0
x2cos xydx
Exercise 2.2 Evaluate
a) I (y) =
1
R
0
arctanx
ydx b) J (y) =
1
R
0
ln x2+ y2 dx c) K =
1
R
0
xb−x a
ln x , (0 < a < b)
2.2 Improper Integrals depending on a parameter
Exercise 2.3 Show that the integral
a) I(y) =
∞
R
1
sin(yx)dx is convergent if y = 0 and is divergent if y 6= 0
Trang 6b) I(y) = R
0
cos αx
x 2 +1 is uniformly convergent on R
c) I(y) =
1
R
0
x−ydx =
∞
R
1
ty−2dt is convergent if y < 1 and divergent if y ≥ 1
d) I(y) =
+∞
R
0
e−yx sin xx is uniformly convergent on [0, +∞)
e) I(y) =
∞
R
0
cos αx
x 2 +1 is uniformly convergent on R
Exercise 2.4 a) Evaluate I(y) =
+∞
R
0
ye−yxdx (y > 0)
b) Prove that I(y) converges to 1 uniformly on [y0, +∞) for all y0> 0
c) Explain why I(y) is not uniformly convergent on (0, +∞)
Exercise 2.5 Prove that
a)
∞
R
0
e−x2dx =
√ π
2 b)
∞
R
0
sin x
x dx = π2
c)
∞
R
0
sin(x2)dx =
∞
R
0
cos(x2)dx = 12pπ
2
d)
+∞
R
0
e−yx sin xx = π2 − arctan y
e)
∞
R
0
sin yx
x(1+x 2 )dx = π2(1 − e−y), y ≥ 0
f)
∞
R
0
1−cos yx
x 2 =π2|y|
g)
∞
R
0
x sin yx
a 2 +x 2dx = π2e−ay, a, y ≥ 0
h)
∞
R
0
e−yx2dx =
√ π
2 √
y, y > 0
i)
+∞
R
0
e−x2a − e−x2b
dx =√
πb −√
πa, (a, b > 0)
j)
+∞
R
0
arctan x −arctan x
b
x dx = π2lnab, (a, b > 0)
k) lim
y→0 +
+∞
R
0
ye−yxdx
6=
+∞
R
0
lim
y→0 +ye−yx
dx and explain why?
Exercise 2.6 Evaluate (a, b, α, β > 0):
a)
+∞
R
0
e−αx−e−βx
b)
+∞
R
0
e−αx2−e−βx2
c)
+∞
R
0
dx
(x 2 +y)n+1
d)
+∞
R
0
e−ax sin bx−sin cxx
e)
+∞
R
0
e−ax cos bx−cos cxx , (a > 0)
f)
+∞
R
0
e−axcos yx
g)
+∞
R
0
e−x2cos (yx) dx
h)
+∞
R
−∞
arctan(x+y) 1+x 2 dx
i)
+∞
R
0
e−ax2−e −bx2
x dx, where a, b > 0
+∞
R
0
e−ax3−e−bx3
x dx, where a, b > 0
j)
∞
R
0
e−ax2−cos bx
x 2 dx, (a > 0) k)
π
R
0
ln(1 + y cos x)dx,
l)
∞
R
0
e−x2sin axdx,
m)
∞
R
0 sin xy
x dx, y ≥ 0,
Trang 7n) R
0
e−ax2cos bxdx (a > 0),
o)
∞
R
0
x2ne−x2cos bxdx, n ∈ N
p) R
0
sin ax cos bx
q)
∞
R
0
sin ax sin bx
2.3 Euler Integral
Exercise 2.7 Evaluate
a)
π
2
R
0
sin6x cos4xdx
b)
a
R
0
x2n√
a2− x2dx (a > 0)
c)
+∞
R
0
x10e−x2dx
d)
+∞
R
0
√
x
(1+x 2 )2dx
e)
+∞
R
0
1 1+x 3dx
f)
+∞
R
0
x n+1
(1+x n )dx, (2 < n ∈ N)
g)
1
R
0
1
n
√ 1−x ndx, n ∈ N∗ h)
+∞
R
0
x4
(1 + x3)2dx,
3.1 Line Integrals of scalar Fields
Exercise 3.1 Evaluate
a) R
C
(x − y) ds, where C is the circle x2+ y2= 2x
b) R
C
y2ds, where C is the curve
x = a (t − sin t)
y = a (1 − cos t)
, 0 ≤ t ≤ 2π, a > 0
c) R
C
p
x2+ y2ds, where C is the curve
x = (cos t + t sin t)
y = (sin t − t cos t)
, 0 ≤ t ≤ 2π
d) R
C
(x + y)ds, where C is the circle x2+ y2= 2y
e) R
L
xyds, where L is the part of the ellipse xa22 +yb22 = 1, x ≥ 0, y ≥ 0
f) I =R
L
|y|ds, where L is the Cardioid curve r = a(1 + cos ϕ) (a > 0)
g) I =R
L
|y|ds, where L is the Lemniscate curve (x2+ y2)2= a2(x2− y2)
3.2 Line Integrals of vector Fields
Exercise 3.2 Evaluate R
ABCA
2 x2+ y2 dx + x (4y + 3) dy, where ABCA is the quadrangular curve, A(0, 0), B(1, 1), C(0, 2)
Exercise 3.3 Evaluate R
ABCDA
dx+dy
|x|+|y|, where ABCDA is the triangular curve, A(1, 0), B(0, 1), C(−1, 0), D(0, −1)
Trang 83.2.1 Green’s Theorem
Exercise 3.4 Evaluate the integral R
C
(xy + x + y) dx + (xy + x − y) dy, where C is the positively ori-ented circle x2+ y2= R2 by
i) computing it directly and
ii) Green’s Theorem, then compare the results,
Exercise 3.5 Evaluate the following integrals, where C is a half the circle x2+ y2 = 2x, traced from
O(0, 0) to A(2, 0)
a) R
C
(xy + x + y) dx + (xy + x − y) dy
b) R
C
x2 y +x4 dy − y2 x +y4 dx
c) R
C
(xy + exsin x + x + y) dx − (xy − e−y+ x − sin y) dy
Exercise 3.6 Evaluate H
OABO
ex[(1 − cos y) dx − (y − sin y) dy], where OABO is the triangle, O(0, 0), A(1, 1), B(0, 2)
3.2.2 Applications of Line Integrals
Exercise 3.7 Find the area of the domain bounded by an arch of the cycloid
x = a(θ − sin θ)
y = a(1 − cos θ)
and
Ox (a > 0)
3.2.3 Independence of Path
Exercise 3.8 Evaluate
(3,0)
R
(−2,1)
x4+ 4xy3 dx + 6x2y2− 5y4 dy
Exercise 3.9 Evaluate
(2,π)
R
(1,π)
1 −yx22cosxydx + sinyx+yxcosyx dy
4.1 Surface Integrals of scalar Fields
Exercise 4.1 EvaluateRR
S
z + 2x +4y3 dS, where S = (x, y, z) |x
2+y3 +z4 = 1, x, y, z ≥ 0
Exercise 4.2 EvaluateRR
S
x2+ y2 dS, where S = (x, y, z) |z = x2+ y2, 0 ≤ z ≤ 1
Exercise 4.3 EvaluateRR
S
x2y2zdS, where S is the part of the cone z =px2+ y2 lies below the plane
z = 1
Exercise 4.4 EvaluateRR
S
dS (2 + x + y + z)2, where S is the boundary of the triangular pyramid
x + y + z ≤ 1, x ≥ 0, y ≥ 0, z ≥ 0
Trang 94.2 Surface Integrals of vector Fields
Exercise 4.5 EvaluateRR
S
z x2+ y2 dxdy, where S is a half of the sphere x2+ y2+ z2= 1, z ≥ 0, with the outward normal vector
Exercise 4.6 EvaluateRR
S
ydxdz + z2dxdy, where S is the surface x2+y42+ z2= 1, x ≥ 0, y ≥ 0, z ≥ 0, and is oriented downward
Exercise 4.7 EvaluateRR
S
x2y2zdxdy, where S is the surface x2+ y2+ z2 = R2, z ≤ 0 and is oriented upward
4.2.1 The Divergence Theorem
Exercise 4.8 Evaluate the following integrals, where S is the surface x2+ y2+ z2 = a2 with outward orientation
a RR
S
S
x3dydz + y3dzdx + z3dxdy
Exercise 4.9 Evaluate RR
S
y2zdxdy + xzdydz + x2ydxdz, where S is the boundary of the domain x ≥
0, y ≥ 0, x2+ y2≤ 1, 0 ≤ z ≤ x2+ y2 which is outward oriented
Exercise 4.10 Evaluate RR
S
xdydz + ydzdx + zdxdy, where S the boundary of the domain (z − 1)2 ≤
x2+ y2, a ≤ z ≤ 1, a > 0 which is outward oriented
4.2.2 Stokes’ Theorem
Exercise 4.11 Use Stokes’ Theorem to evaluate R
C
F · dr = R
C
P dx + Qdy + Rdz In each case C is oriented counterclockwise as viewed from above
1 F (x, y, z) = (x + y2)i + (y + z2)j + (z + x2)k, C is the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1)
2 F (x, y, z) = i + (x + yz)k + (xy −√
z)k, C is the boundary of the part of the plane 3x + 2y + z = 1
in the first octant
3 F (x, y, z) = yzi + 2xzj + exyk, C is the circle x2+ y2= 16, z = 5
4 F (x, y, z) = xyi + 2zj + 3yk, C is the curve of intersection of the plane x + z = 5 and the cylinder
y2+ y2= 9
5.1 Scalar Fields
Exercise 5.1 Find the directional derivative of the function f (x, y, z) = x2y3z4 at the point M (1, 1, 1)
in the direction of the vector ~l = (1, 1, 1)
Trang 10Exercise 5.2 FindOu, where u = r +r + ln r and r = x2+ y2+ z2.
Exercise 5.3 In what direction from O(0, 0, 0) does f = x sin z − y cos z have the maximum rate of change
5.2 Vector Fields
Exercise 5.4 Let F = xz2−→
i + yx2−→
j + zy2−→
k Find the flux of F across the surface S : x2+ y2+ z2= 1 with the outward direction
Exercise 5.5 Let F = x(y + z)−→
i + y(z + x)−→
j + z(x + y)−→
k and L is the intersection between the quatity
x2+ y2+ y = 0 and a half of the sphere x2+ y2+ z2= 2, z ≥ 0 Prove that the circulation of F across L
is equal to 0
Exercise 5.6 Prove that F is a conservative vector field on Ω if and only if curl F (M ) = 0 ∀M ∈ Ω Exercise 5.7 Which of the following fields are conservative and find their potential functions
a F = 5(x2− 4xy)−→i + (3x2− 2y)−→j +−→
k
b G = yz−→
i + xz−→
j + xy−→
k
c H = (x + y)−→
i + (x + z)−→
j + (z + y)−→
k
6.1 Infinite series
Exercise 6.1 Show that the harmonic series
∞
P
n=1
1
n is divergent
Exercise 6.2 Find the sum of the series
∞
P
n=1
3 n(n+1) +21n
Exercise 6.3 Test for convergence or divergence of the series
a)
∞
P
n=1
∞
P
n=1
∞
P
n=1
n n+1
n
6.1.1 The Integral Test
Exercise 6.4 Show that the series
∞
P
n=2
1 n(ln n) p is convergent iff p > 1
Exercise 6.5 Test for convergence or divergence of the series
a)
∞
P
n=1
ln 1
n
(n+2) 2
b)
∞
P
n=1
n2e−n3
c)
∞
P
n=1
ln n
n 3
d)
∞
P
n=1
ln(1+n) (n+3) 2
e)
∞
P
n=1
e1/n
n 2
f)
∞
P
n=1
n2
e n
g)
∞
P
n=1
ln n
n p
h)
∞
P
n=1
ln n 3n 2
i)
∞
P
n=1
1 ln(2n+1)
j)
∞
P
n=2
1 ln(2n−1)
k)
∞
P
n=1
cos n
√
n 3 +1
l)
∞
P
n=1
sin n
√
n 3 +1
Trang 116.1.2 The Comparison Test
Exercise 6.6 Test for convergence or divergence of the series
1)
∞
P
n=1
n3
(n+2) 4
2)
∞
P
n=1
2016n
2015 n +2017 n
3)
∞
P
n=1
n sin 2 n
1+n 3
4)
∞
P
n=1
3
√
n
√
n+3
5)
∞
P
n=1
sin(√
n + 1 −√
n)
6)
∞
P
n=1
n+sin n
3
√
n 7 +1
7)
∞
P
n=1
sinn3n+1+n+1
8)
∞
P
n=1
ln1 + 1
3n 2
9)
∞
P
n=1
1
ln(2n+1)
10)
∞
P
n=1
cos n
√
n 3 +1
11)
∞
P
n=1
1
n − sin1 n
12)
∞
P
n=1
1 − cos1n
13)
∞
P
n=1
n
√
e − 1 −n1
14)
∞
P
n=1
arcsinn2n−1−n+1
15)
∞
P
n=2
1 [ln(ln(n+1))] ln n
16)
∞
P
n=1
nen1 − 12,
17)
∞
P
n=2
(−1) n +1 n−ln n , 18)
∞
P
n=1
arcsin(e−n),
19)
∞
P
n=1
sin(π√
n2+ a2),
20)
∞
P
n=1
(2n−1)!!
3 n n! , 21)
∞
P
n=1
cosann
3
,
22)
∞
P
n=1
n n2 2 n
(n+1) n2, 23)
∞
P
n=3
1
n α (ln n) β, (α, β > 0), 24)
∞
P
n=3
(−1)n+2 cos nα n(ln n)3 , 25)
∞
P
n=1
na (1−a 2 ) n, 0 < |a| 6= 1 26)
∞
P
n=1
(n!) 2
4 n2 , 27)
∞
P
n=1
cos 1 n+1− cos1
n
6.1.3 Alternating Series
Exercise 6.7 Test for convergence or divergence of the following series
a)
∞
P
n=1
(−1)n−1
n+1
b)
∞
P
n=1
(−1)n−1 n2
n 3 +1 c)
∞
P
n=1
(−1)n 2n+1
3n+2n,
d)
∞
P
n=1
(−1)nnn3 +42 ,
e)
∞
P
n=1
(−1) n (n 2 +n+1)
2 n (n+1) , f)
∞
P
n=1
(−1)nsin πn,
g)
∞
P
n=1
(−1)nn2
π n , h)
∞
P
n=1
(−1) n
3 n n!, i)
∞
P
n=1
(−1)nn+1
n+2
n
,
j)
∞
P
n=1
(−1)nsinn√1
n, k)
∞
P
n=1
(−1)n ln n
n
6.1.4 The ratio (d’Alambert) Test
Exercise 6.8 Test for convergence or divergence of the series
a)
∞
P
n=1
2n
n!
b)
∞
P
n=1
2nn!
n n
c)
∞
P
n=1
5 n (n!) 2
n 2n
d)
∞
P
n=1
(2n+1)!!
n n
e)
∞
P
n=1
(n 2 +n+1)
2 n (n+1)
f)
∞
P
n=1
(2n)!!
n n
g)
∞
P
n=1
22n+1
5 n ln(n+1)
h)
∞
P
n=1
lnh1 + 2n+1n +1
i
6.1.5 The root (Cauchy) Test
Exercise 6.9 Test for convergence or divergence of the series