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Hanoi University of Science and Technology Dr Bui Xuan Dieu School of Applied Mathematics and Informatics Class ICT K62 Math4 Exercises Multiple Integrals 1.1 Double Integrals 1.1.1 Double Integrals in Cartesian coordinate Exercise 1.1 Evaluate π 2,0 x sin(x + y)dxdy, where D = (x, y) ∈ R2 : ≤ y ≤ a) D ≤x≤ π x2 (y − x) dxdy where D is the region bounded by y = x2 and x = y b) D |x + y|dxdy, D := (x, y) ∈ R2 ||x ≤ 1| , |y| ≤ c) D |y − x2 |dxdy, D := (x, y) ∈ R2 ||x| ≤ 1, ≤ y ≤ d) D e) ydxdy 2 [0,1]×[0,1] (1+x +y ) x2 y dxdy, f) D 1.1.2 where D is bounded by the lines x = 2, y = x and the hyperbola xy = Change the order of integration Exercise 1.2 Change the order of integration 1−x2 dx a) −1 √ 1+ 0 √ 1−y √ f (x, y) dx √ y 2−y 2x 2x−x2 dy d) f (x, y) dx dy b) dx c) − 1−x2 1.1.3 f (x, y) dy √ √ 2 f (x, y) dx+ √ Change of variables 4x2 − 2y dxdy, where D : Exercise 1.3 Evaluate I = D   1 ≤ xy ≤  x ≤ y ≤ 4x Exercise 1.4 Evaluate x2 sin xy dxdy, y I= D where D is bounded by parabolas x2 = ay, x2 = by, y = px, y = qx, (0 < a < b, < p < q) 4−y dy f (x, y) dx Exercise 1.5 Evaluate I = xydxdy, where D is bounded by the curves D y = ax3 , y = bx3 , y = px, y = qx, (0 < b < a, < p < q) Hint: Change of variables u = x3 y ,v y2 x = Exercise 1.6 Prove that 1−x e−1 y dx e x+y dy = Hint: Change of variables u = x + y, v = y Exercise 1.7 Find the area of the domain bounded by xy = 4, xy = 8, xy = 5, xy = 15 Hint: Change of variables u = xy, v = xy , (S = ln 3) Exercise 1.8 Find the area of the domain bounded by y = x, y = 8x, x2 = y, x2 = 8y Hint: Change of variables u = y2 x ,v = x2 y , 279π ) (S = Exercise 1.9 Hint: Change of variables y = x3 , y = 4x3 , x = y , x = 4y Exercise 1.10 Prove that x−y x+y cos dxdy = sin x+y≤1,x≥0,y≥0 Hint: Change of variables u = x − y, v = x + y Exercise 1.11 Evaluate x + a I= y b dxdy, D x a where D is bounded by the axes and the parabola 1.1.4 + y b = Double Integrals in polar coordinate f (x, y) dxdy in terms of polar coordinates, where D Exercise 1.12 Express the double integral I = D√ is given by x2 + y ≥ 4x, x2 + y ≤ 8x, y ≥ x, y ≤ 3x xy dxdym where D is bounded by Exercise 1.13 Evaluate   x2 + (y − 1)2 =  x2 + y − 4y = D Exercise 1.14 Evaluate |x + y|dxdy, a) |x − y|dxdy, b) D D where D : x2 + y ≤ Exercise 1.15 Evaluate D Exercise 1.16 Evaluate D   4y ≤ x2 + y ≤ 8y dxdy , where D : (x2 +y )2 √  x ≤ y ≤ x   x2 + y ≤ 12, x2 + y ≥ 2x xy dxdy, where D : 2 x +y √  x2 + y ≥ 3y, x ≥ 0, y ≥ 1.2 Applications of Double Integrals Exercise 1.17 Compute the area of the domain D bounded by a) b) c)   y = 2x , y = 2−x , d)  y =   y = x, y = 2x   x2 + y = 2x, x2 + y = 4x  x = y, y = e) r = 1, r = f) x2 + y 2 √2 cos ϕ = 2a2 xy (a > 0)  x2 = y, x2 = 2y   y = 0, y = 4ax g) x3 + y = axy (a > 0) (Descartes leaf)  x + y = 3a, (a > 0) h) r = a (1 + cos ϕ) (a > 0) (Cardioids) Exercise 1.18 Compute the volume of the object given by   ≤ z ≤ − x2 − y b) V : √  y ≥ x, y ≤ 3x    3x + y ≥ 1, y ≥    a) 3x + 2y ≤ 2,     0 ≤ z ≤ − x − y c) V :   x2 + y + z ≤ 4a2  x2 + y − 2ay ≤ Exercise 1.19 Compute the volume of the object bounded by the surfaces a)  x2 y2   z = + ,z = a b b) 2  y 2x x   + = a b a   z = − x − y  2z = + x2 + y c)   az = x2 + y  z= x2 + y Exercise 1.20 Find the area of the part of the paraboloid x = y + z that satisfies x ≤ 1.3 Triple Integrals 1.3.1 Triple Integrals in Cartesian coordinate x2 + y dxdydz, where V is bounded by the sphere x2 + y + z = and Exercise 1.21 Evaluate V the cone x2 + y − z = 1.3.2 Change of variables Exercise 1.22 Evaluate a) V    x + y + z = ±3    (x + y + z)dxdydz, where V is bounded by x + 2y − z = ±1      x + 4y + z = ±2 (3x2 + 2y + z)dxdydz, where V : |x − y| ≤ 1, |y − z| ≤ 1, |z + x| ≤ b) V dxdydz, where V : |x − y| + |x + 3y| + |x + y + z| ≤ c) V 1.3.3 Triple Integrals in Cylindrical Coordinates x2 + y dxdydz, where V : Exercise 1.23 Evaluate V  1≤z≤2 x2 + y dxdydz, where: z Exercise 1.24 Evaluate   x2 + y ≤ V a) V is bounded by: x2 + y = 2x and z = 0, z = a (a > 0) b) V is a half of the sphere x2 + y + z ≤ a2 , z ≥ (a > 0) x2 + y dxdydz where V is bounded by: Exercise 1.25 Evaluate I = V √ Exercise 1.26 Evaluate V 1.3.4   x2 + y = z  dxdydz , x2 +y +(z−2)2 where V : z =   x2 + y ≤  |z| ≤ Triple Integrals in Spherical Coordinates x2 + y + z dxdydz, where V : Exercise 1.27 Evaluate V   ≤ x2 + y + z ≤  x2 + y ≤ z x2 + y + z dxdydz, where V : x2 + y + z ≤ z Exercise 1.28 Evaluate V x2 + y dxdydz, where V is a half of the ellipsoid z Exercise 1.29 Evaluate V x2 +y a2 + z2 b2 ≤ 1, z ≥ 0, (a, b > 0) Exercise 1.30 Evaluate V x2 a2 + y2 b2 + z2 c2 dxdydz , where V : x2 a2 + y2 b2 + z2 c2 ≤ 1, (a, b, c > 0) z − x2 − y − z dxdydz, where V : x2 + y + z ≤ z Exercise 1.31 Evaluate V (4z − x2 − y − z )dxdydz, where V is the sphere x2 + y + z ≤ 4z Exercise 1.32 Evaluate V xzdxdydz, where V is the domain x2 + y + z − 2x − 2y − 2z ≤ −2 Exercise 1.33 Evaluate V Exercise 1.34 Evaluate dxdydz , (1 + x + y + z)3 I= V where V is bounded by x = 0, y = 0, z = x + y + z = Exercise 1.35 Evaluate zdxdydz, V where V is a half of the ellipsoid x2 y2 z2 + + ≤ 1, (z ≥ 0) a2 b2 a2 Exercise 1.36 Evaluate a) I1 = B x2 a2 + y2 b2 + z2 c2 , where B is the ellipsoid x2 a2 + y2 b2 + z2 c2 ≤ zdxdydz, where C is the domain bounded by the cone z = b) I2 = C h2 R2 (x + y ) and the plane z = h z dxdydz, where D is bounded by the sphere x2 +y +z ≤ R2 and the sphere x2 +y +z ≤ c) I3 = D 2Rz (x + y + z)2 dxdydz, where V is bounded by the paraboloid x2 + y ≤ 2az and the sphere d) I4 = V x2 + y + z ≤ 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= y2 x2 + , (p > 0, q > 0) 2p 2y Exercise 1.38 Evaluate x2 + y + z dxdydz, I= V 2 where V is the domain bounged by x + y + z = z Exercise 1.39 Evaluate I= zdxdydz, V where V is the domain bounded by the surfaces z = x2 + y and x2 + y + z = Exercise 1.40 Evaluate xyz dxdydz, + y2 I= x2 V where V is the domain bounded by the surface (x2 + y + z )2 = a2 xy and the plane z = Integrals depending on a parameter 2.1 Definite Integrals depending on a parameter Exercise 2.1 Compute 1+y a) lim y→0 y dx 1+x2 +y b) lim y→0 x2 cos xydx Exercise 2.2 Evaluate a) I (y) = 2.2 arctan xy dx 1 ln x2 + y dx b) J(y) = 0 Improper Integrals depending on a parameter Exercise 2.3 Show that the integral ∞ a) I(y) = c) K = sin(yx)dx is convergent if y = and is divergent if y = xb −xa ln x , (0 < a < b) ∞ b) I(y) = c) I(y) = cos αx x2 +1 is uniformly convergent on R x−y dx = +∞ ∞ ty−2 dt is convergent if y < and divergent if y ≥ 1 d) I(y) = e) I(y) = ∞ e−yx sinx x is uniformly convergent on [0, +∞) cos αx x2 +1 is uniformly convergent on R +∞ a) Evaluate I(y) = Exercise 2.4 ye−yx dx (y > 0) b) Prove that I(y) converges to uniformly on [y0 , +∞) for all y0 > c) Explain why I(y) is not uniformly convergent on (0, +∞) Exercise 2.5 Prove that ∞ a) e−x dx = ∞ b) π = g) cos(x2 )dx = +∞ d) 0 ∞ ∞ ∞ π sin(x2 )dx = e) f) ∞ sin x x dx ∞ c) ∞ √ π h) 1−cos yx x2 x sin yx a2 +x2 dx sin yx x(1+x2 ) dx +∞ k) lim+ y→0 = π 2 π (1 π −ay , 2e √ π √ y, a, y ≥ y > √ b a √ e− x2 − e− x2 dx = πb − πa, (a, b > 0) i) − arctan y π |y| = e−yx dx = +∞ e−yx sinx x = = − e−y ), ye−yx dx +∞ y ≥ x x a −arctan b arctan j) x +∞ = dx = π ln ab , (a, b > 0) lim ye−yx dx and explain why? y→0+ Exercise 2.6 Evaluate (a, b, α, β > 0): +∞ a) +∞ b) +∞ c) +∞ d) +∞ e) +∞ f) +∞ e−αx −e−βx dx x e−αx −e−βx x2 h) −∞ +∞ dx i) +∞ dx (x2 +y)n+1 ∞ cx e−ax sin bx−sin x j) cx e−ax cos bx−cos , (a > 0) x k) e−ax cos yx l) e−ax −e−bx x e−ax −e−bx x dx, e−ax −cos bx dx, (a x2 ln(1 + y cos x)dx, 0 ∞ e−x cos (yx) dx m) 0 e−x sin axdx, sin xy x dx, y a, b dx, where a, b > 0 ∞ where π +∞ g) arctan(x+y) dx 1+x2 ≥ 0, > 0) > ∞ n) ∞ e−ax cos bxdx (a > 0), p) 0 ∞ o) ∞ x2n e−x cos bxdx, n ∈ N q) 0 2.3 sin ax cos bx dx, x sin ax sin bx dx x Euler Integral Exercise 2.7 Evaluate π +∞ sin6 x cos4 xdx a) e) 0 √ x2n a2 − x2 dx (a > 0) a b) +∞ f) 0 +∞ c) x10 e−x dx g) 0 +∞ d) √ h) xn+1 (1+xn ) dx, √ dx, n 1−xn +∞ x dx (1+x2 )2 1+x3 dx (2 < n ∈ N) n ∈ N∗ x4 dx, (1 + x3 )2 Line Integrals 3.1 Line Integrals of scalar Fields Exercise 3.1 Evaluate (x − y) ds, where C is the circle x2 + y = 2x a) C y ds, where C is the curve b) C   x = a (t − sin t) , ≤ t ≤ 2π, a >  y = a (1 − cos t) x2 + y ds, where C is the curve c) C   x = (cos t + t sin t) , ≤ t ≤ 2π  y = (sin t − t cos t) (x + y)ds, where C is the circle x2 + y = 2y d) C xyds, where L is the part of the ellipse e) L x2 a2 + y2 b2 = 1, x ≥ 0, y ≥ |y|ds, where L is the Cardioid curve r = a(1 + cos ϕ) (a > 0) f) I = L |y|ds, where L is the Lemniscate curve (x2 + y )2 = a2 (x2 − y ) g) I = L 3.2 Line Integrals of vector Fields x2 + y dx + x (4y + 3) dy, where ABCA is the quadrangular curve, Exercise 3.2 Evaluate ABCA A(0, 0), B(1, 1), C(0, 2) Exercise 3.3 Evaluate ABCDA dx+dy |x|+|y| , where ABCDA is the triangular curve, A(1, 0), B(0, 1), C(−1, 0), D(0, −1) 3.2.1 Green’s Theorem (xy + x + y) dx + (xy + x − y) dy, where C is the positively ori- Exercise 3.4 Evaluate the integral C ented circle x2 + y = 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 + y = 2x, traced from O(0, 0) to A(2, 0) (xy + x + y) dx + (xy + x − y) dy a) C x2 y + b) C x dy − y x + y dx (xy + ex sin x + x + y) dx − (xy − e−y + x − sin y) dy c) C ex [(1 − cos y) dx − (y − sin y) dy], where OABO is the triangle, O(0, 0), A(1, 1), B(0, 2) Exercise 3.6 Evaluate OABO 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) and  y = a(1 − cos θ) Ox (a > 0) 3.2.3 Independence of Path (3,0) x4 + 4xy dx + 6x2 y − 5y dy Exercise 3.8 Evaluate (−2,1) (2,π) 1− Exercise 3.9 Evaluate y2 x2 cos xy dx + sin xy + y x cos xy dy (1,π) Surface Integrals 4.1 Surface Integrals of scalar Fields z + 2x + Exercise 4.1 Evaluate S 4y dS, where S = (x, y, z) | x2 + y + z = 1, x, y, z ≥ x2 + y dS, where S = (x, y, z) |z = x2 + y , ≤ z ≤ Exercise 4.2 Evaluate S x2 y zdS, where S is the part of the cone z = Exercise 4.3 Evaluate x2 + y lies below the plane S z = dS , where S is the boundary of the triangular pyramid (2 + x + y + z)2 S x + y + z ≤ 1, x ≥ 0, y ≥ 0, z ≥ Exercise 4.4 Evaluate 4.2 Surface Integrals of vector Fields z x2 + y dxdy, where S is a half of the sphere x2 + y + z = 1, z ≥ 0, with Exercise 4.5 Evaluate S the outward normal vector ydxdz + z dxdy, where S is the surface x2 + y4 + z = 1, x ≥ 0, y ≥ 0, z ≥ 0, Exercise 4.6 Evaluate S and is oriented downward x2 y zdxdy, where S is the surface x2 + y + z = R2 , z ≤ and is oriented Exercise 4.7 Evaluate S upward 4.2.1 The Divergence Theorem Exercise 4.8 Evaluate the following integrals, where S is the surface x2 + y + z = a2 with outward orientation xdydz + ydzdx + zdxdy a x3 dydz + y dzdx + z dxdy b S S y zdxdy + xzdydz + x2 ydxdz, where S is the boundary of the domain x ≥ Exercise 4.9 Evaluate S 0, y ≥ 0, x2 + y ≤ 1, ≤ z ≤ x2 + y which is outward oriented xdydz + ydzdx + zdxdy, where S the boundary of the domain (z − 1) ≤ Exercise 4.10 Evaluate S x2 + y , a ≤ z ≤ 1, a > which is outward oriented 4.2.2 Stokes’ Theorem F · dr = Exercise 4.11 Use Stokes’ Theorem to evaluate C P dx + Qdy + Rdz In each case C is C oriented counterclockwise as viewed from above F (x, y, z) = (x + y )i + (y + z )j + (z + x2 )k, C is the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) √ F (x, y, z) = i + (x + yz)k + (xy − z)k, C is the boundary of the part of the plane 3x + 2y + z = in the first octant F (x, y, z) = yzi + 2xzj + exy k, C is the circle x2 + y = 16, z = F (x, y, z) = xyi + 2zj + 3yk, C is the curve of intersection of the plane x + z = and the cylinder y + y = 5.1 Vector Calculus Scalar Fields Exercise 5.1 Find the directional derivative of the function f (x, y, z) = x2 y z at the point M (1, 1, 1) in the direction of the vector l = (1, 1, 1) Exercise 5.2 Find u, where u = r2 + r x2 + y + z + ln r and r = 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 = xz i + yx2 j + zy k Find the flux of F across the surface S : x2 + y + z = 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 + y + y = and a half of the sphere x2 + y + z = 2, z ≥ Prove that the circulation of F across L is equal to Exercise 5.6 Prove that F is a conservative vector field on Ω if and only if curl F (M ) = ∀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 Series 6.1 Infinite series ∞ Exercise 6.1 Show that the harmonic series n=1 ∞ Exercise 6.2 Find the sum of the series n=1 n is divergent n(n+1) + 2n Exercise 6.3 Test for convergence or divergence of the series ∞ a) n=1 6.1.1 ∞ n sin n+sin 3n+1 b) n=1 ∞ cos n1 c) n=1 n n+1 n The Integral Test ∞ Exercise 6.4 Show that the series n=2 n(ln n)p is convergent iff p > Exercise 6.5 Test for convergence or divergence of the series ∞ a) n=1 ∞ b) ∞ ln n (n+2)2 n2 e−n d) n=1 ∞ e) n=1 ∞ c) n=1 n=1 ln n n3 ∞ f) n=1 ∞ ln(1+n) (n+3)2 g) e1/n n2 h) n2 en i) n=1 ∞ n=1 ∞ n=1 10 ∞ ln n np j) ln n 3n2 k) ln(2n+1) l) n=2 ∞ n=1 ∞ n=1 ln(2n−1) √cos n n3 +1 √sin n n3 +1 6.1.2 The Comparison Test Exercise 6.6 Test for convergence or divergence of the series ∞ 1) n=1 ∞ 2) n=1 ∞ 3) n=1 ∞ 4) n=1 ∞ n3 (n+2)4 10) 2016n 2015n +2017n 11) n sin2 n 1+n3 12) √ √ n n+3 13) n=1 ∞ ∞ n=1 ∞ ∞ 14) n=1 ∞ n=1 ∞ 7) n=1 15) sin n3n+1 +n+1 16) 3n2 17) ∞ ln + 8) n=1 ∞ 9) n=1 6.1.3 − cos n1 21) e−1− n=1 ∞ n=1 ∞ n 22) n=1 ∞ arcsin n2n−1 −n+1 23) [ln(ln(n+1))]ln n 24) ∞ n+sin n √ n +1 ∞ 20) √ n n=2 ∞ n=3 arcsin(e−n ), 27) n=2 ∞ ln(2n+1) 18) n=1 ∞ n=1 nn 2n (n+1)n2 , , (α, β nα (ln n)β ∞ 26) , n(ln n) n=1 (−1)n +1 n−ln n , n3 n=3 ∞ n=1 cos na (−1)n +2 cos nα 25) , (2n−1)!! 3n n! , ∞ n en − ∞ √ sin(π n2 + a2 ), n=1 n=1 n=1 6) 19) − sin n1 n n=1 √ √ sin( n + − n) ∞ 5) ∞ √cos n n3 +1 n=1 na (1−a2 )n , > 0), , < |a| = (n!)2 , 4n2 cos n+1 − cos n1 Alternating Series Exercise 6.7 Test for convergence or divergence of the following series ∞ a) n=1 ∞ b) ∞ (−1)n−1 n+1 d) n=1 ∞ (−1)n−1 n3n+1 e) 2n+1 (−1)n 3n+2n , f) n=1 ∞ c) n=1 ∞ n=1 6.1.4 ∞ (−1)n n3n+4 , g) n=1 ∞ (−1)n (n2 +n+1) , 2n (n+1) (−1)n sin n=1 π n h) n=1 ∞ i) , ∞ (−1)n n2 , πn j) (−1)n 3n n! , k) (−1)n n=1 n=1 ∞ n=1 n+1 n+2 , (−1)n sin n√ n (−1)n lnnn n , The ratio (d’Alambert) Test Exercise 6.8 Test for convergence or divergence of the series ∞ a) n=1 ∞ b) n=1 6.1.5 ∞ 2n n! c) 2n n! nn d) n=1 ∞ n=1 ∞ 5n (n!)2 n2n e) (2n+1)!! nn f) n=1 ∞ n=1 The root (Cauchy) Test Exercise 6.9 Test for convergence or divergence of the series 11 ∞ (n2 +n+1) 2n (n+1) g) (2n)!! nn h) n=1 22n+1 5n ln(n+1) ∞ ln + n=1 n+1 2n +1 ∞ n2 +n+1 3n2 +n+1 a) n=1 ∞ n n+2 b) n=1 n ∞ n c) n=1 ∞ nn 5n ∞ n=1 n=1 ∞ n(n+4) n+2 n+3 d) n+3 n+2 e) 2n (n+1)n2 2n+1 3n+1 f) n=1 ∞ n(n+4) g) n=1 ∞ n h) n=1 √ n2 + n+sin n 2n2 +1 n n+1 n2 Exercise 6.10 Test for convergence or divergence of the series ∞ (a) n=1 ∞ √ (b) n=2 ∞ n=2 ∞ √ ∞ √1 n (l) n=2 ∞ n=2 ∞ (k) (g) (h) n=1 ln n √ , n n=2 , (i) (j) ∞ n4 ∞ , ln n , (f) , √ n+1− n−1 n=1 6.1.6 n=1 1+n n n n2 (e) n , (n−1)(n+2) 1+n n2 −1 (c) (d) ∞ n 10n2 +1 , n=2 ∞ n=1 ∞ 1+n ln n−1 , n=2 n − ln 1+n , n √ ln nn2+−nn tan n12 , (3n+1)! n2 8n , 1.3.5 (2n−1) 22n (n−1)! Absolute and Conditional Convergence Exercise 6.11 Test for absolute or conditional convergence of the series ∞ a) n=1 ∞ b) ∞ sin √ n n3 d) n=1 2n+1 (−1)n 3n+2n ∞ e) n=1 ∞ c) (−1)n n3n+4 (−1)n sin n=1 ∞ f) n=1 n=1 ∞ (−1)n (n2 +n+1) 2n (n+1) n (−1) n πn ∞ Exercise 6.12 Prove that the series n=1 ∞ Exercise 6.13 Prove that the series n=1 g) n=1 ∞ π n h) ∞ i) 6.2.1 j) n=1 n+1 n+2 (−1)n lnnn n (−1)n sin n√ n n=1 sin n n is a conditionally convergent sin n np is b) conditionally convergent if < p ≤ a) absolutely convergent if p > 1, 6.2 (−1)n n=1 ∞ (−1)n 3n n! Series of Functions Domain of convergence Exercise 6.14 Find the domain of convergence of the series ∞ a) ∞ c) n=1 n=1 ∞ ∞ b) n=1 6.2.2 xn nx d) n=1 ∞ sin x+cos x n2 +x2 e) n=1 ∞ xn n! f) n=1 g) (2n)!! n nn x h) Uniform convergence Exercise 6.15 Test for uniform convergence of the following series 12 ∞ sin nx 2n (n+1) n=1 ∞ n=1 22n+1 xn 5n x sin n+sin 3n+1 3n ∞ a) n=1 ∞ b) n=1 (−1)n−1 x2 +n2 , x sin nx n2 +x2 , x ∞ ∈ R c) n=1 ∞ ∈ R d) n=1 xn√ ,x 2n n n 2n−1 ∞ Exercise 6.16 Test for continuity of the series of functions n=1 n2 ∈ [−2, 2] 2x+1 x+2 n , x ∈ [−1, 1] x arctan √n+1 Exercise 6.17 Find the domain of convergence and its sum ∞ a) ∞ (−1)n−1 (n + 1)(x − 1)n c) n=1 ∞ b) n=1 ∞ (−1)n (2n + 1)x2n d) n=1 n=1 (−1)n−1 (x n + 1)n x2n+1 2n+1 Exercise 6.18 Prove that ∞ a) arctan x = 2n+1 (−1)n x2n+1 = x − n=0 b) 6.3 π ∞ = n=0 x3 + x5 2n+1 − · · · + (−1)n x2n+1 + · · · , x ∈ [−1, 1] (−1)n 2n+1 Power Series Exercise 6.19 Find interval of convergence of the series ∞ a) ∞ 2n x (−1)n (2n)! c) n=0 ∞ b) n=0 n(x+2) 3n+1 n=0 ∞ n d) n=0 ∞ n(x+1)n 4n n (x+4) √ n+1 e) n!(2x − 1)n n=1 ∞ n f) n=2 x2n n(ln n)2 Exercise 6.20 Find a power series representation for a) f (x) = ln(1 + x) f) f (x) = 1−4x2 k) f (x) = sin2 x b) f (x) = ln(2 + x) g) f (x) = 1−x 1+x l) f (x) = ex sin x h) f (x) = x2 −x−2 i) f (x) = x+2 2x2 −x−1 j) f (x) = x2 +x (1−x)3 x c) f (x) = 1+x2 d) f (x) = arctan x m) f (x) = 1+x n) f (x) = ln 1−x x e) f (x) = 6.4 3−x e−t dt o) f (x) = sin t t dt Fourier Series Exercise 6.21 Find the Fourier series of the 2π-periodic function defined as a) f (x) =   1, 0≤x≤π  −1, −π ≤ x < b) f (x) = 13   x, 0≤x≤π  −1, −π ≤ x < c) f (x) = x2 , −π < x < π d) f (x) =   1, 0≤x≤π  0, −π ≤ x < Exercise 6.22 Find the Fourier cosine series and Fourier sine series of the following functions a) f (x) =   1, 0≤x≤  0, π π c) f (x) = π + x, ≤ x ≤ π