Eercises of Mathematical analysis II In eercises. - 8. represent the domain of the function by the inequalities and make a sketch showing the domain in y-plane.. z = y.. z = arcsin y + + ln y. 3. z = sin π( + y ). 4. z = ln ln sin y. 5. z = 4 + ln(y 4) 6. z = arcsin y 7. z = (y + y) cos 8. z = + y ln(9 y ) 9. Are the functions identical? Why?. Are the functions identical? Why? z = sin y and z = sin y z = ln y and z = ln + ln y. f(, y) = 3 y. Evaluate lim + y f(, y) eist? lim (,y) (,) Evaluate the limit + y. lim (,y) (,) + y + lim y f(, y) and lim lim f(, y). oes y
3. lim (,y) (,) y + + y y 4. lim (,y) (,) + y y( y ) 5. lim (,y) (,) + y y 6. lim (,y) (,) + y sin( 3 + y 3 ) 7. lim (,y) (,) + y In eercises 8. - 8. find all first-order partial derivatives of the given function. 8. z = 3 y + 4 y 9. z = ln tan y. z = e y.. z = sin y cos y.. z = ln( + + y ) 3. z = arctan y. 4. z = y ln( + y). 5. w = ln(y + ln z) 6. w = tan( + y 3 + z 4 ). 7. w = yz. 8. z = ( + y ) + y + + y. 9. Evaluate the partial derivatives of z = arcsin at (; ). + y
3. z = cos y y cos z. Evaluate + sin + sin y z and, if = y =. y 3. w = ln( + + y + z 3 ). Evaluate w + w y + w at the point = z y = z = 3. z = ln( y ); prove that z + z y + y = 33. For the function z = y + arctan y z z prove that et +y y 34. Find the total differential of the function z = arcsin y. = y +z. 35. Find the total differential of the function z = sin y cos y. 36. Find the total differential of the function z = ln sin y. 37. Find the total differential of the function w = yz. 38. Find the total differential of the function z = y, if =, y =, y =, and y =, 3. 39. Evaluate the increment z and total differential dz of the function z = + y y, if = 3, y = 7, = 3 and y = 4. 4. Evaluate the increment z and total differential dz of the function z = y +, if changes from to, 8 and y from to,. y 4. Using total differential, compute the approimate value of, 96 3, 3 5. 4. Using total differential, compute the approimate value of 8 3 8. 43. Using total differential, compute the approimate value of arcsin, 4, 4. 44. Using total differential, compute the approimate value of ln ( 5, 98 + 4, 4 ). 45. Find d, if y 4 y 4 = a. 3
46. Find d, if y3 + 3 y + ln = and evaluate it at =. 47. Find d, if y = ln y and evaluate it at the point (e ; e). 48. Find d, if y = y and evaluate it at the point (; ). 49. Find z and z y, if y + z 4 + z 5 =. 5. Find z z and, if z = cos y sin z and evaluate these at the point ( y π ) ; ;. 5. Find z z and y, if yz = ez and evaluate these at the point (e ; ; ). 5. Find the total differential of z, if z is determined by the equation cos + cos y + cos z =. 53. Find dz, if z = arctan(y + ) and y = ln. d 54. Find dz d, if z = arcsin y and y = + 4. 55. Find dz dt, if z = tan(3t + y), = t and y = t. 56. Find du e, if u = (y z), y = sin and z = cos. d 5 57. Find dw d, if w = + u + v, u = sin and v = e. 58. Find dz dt, if z = arcsin y, = sin t and y = t. 59. Find z u 6. Find z u and z v, if z = + y, = u cos v and y = v cos u. and z v, if z = ln( + y ), = u cosh v and y = v sinh u. 6. Find the partial derivatives z if u = y and v = y. and z y 4 of the function z = arctan uv,
6. Prove that z y = 63. Find z, z y and 64. Find 3 w y z, if w = eyz. z y, if z = e (cos y + sin y). z, if z = arcsin(y). y 65. Evaluate all second order derivatives of the function z = y at the point ( ; ). 66. Evaluate all second order derivatives of the function z = arcsin at the point (; ). + y 67. Evaluate ( ) z z y z y at the point (; ), if z = cos. y 68. For the function z = ln (e + e y ) prove that z + z y = and z z y ( ) z =. y 69. Prove that the function z = y + y satisfies the equation z + y z y = z 7. Find the canonical equations of the tangent line of spatial curve = + sin t, y = t cos t, z = 3 + t at t =. 7. Find the canonical equations of the tangent line of screw line = cos t, y = sin t, z = 4t at t = π 4 7. Find the canonical equations of the tangent line of spatial curve = t sin t, y = cos t, z = 4 sin t ( π ) at ; ;. 5
73. On the curve y =, z = 3 find the points at which the tangent line is parallel to the plane + y + z =. 74. Find the equation of the tangent plane and the canonical equations of the normal line for the surface z = arctan y ( at the point ; ; π ). 4 75. Find the equation of the tangent plane and the canonical equations of the normal line for the surface z = + y at (3; 4; 5). 76. Find the equation of the tangent plane and the canonical equations of the normal line for the surface z = cos y at ( ; π; ). 77. Find the equation of the tangent plane and the canonical equations of the normal line for the surface y + + z 3 = 6 at the point = and y =. 78. Prove that the surfaces + y ln z = 4 and y 8 + z = 5 have the same tangent plane at the point (; 3; ). 79. Find the gradient vector for the scalar field z = 3y + 3y at the point (3; 4). 8. Find ( the points at which the gradient vector of the scalar field z = ln + ) is ( a = ; 6 ). y 9 8. Find the gradient vector for the scalar field w = arcsin the point (; ; ). + y z at 8. Find the directional derivative of the function z = arctan y 4y point (; 3) in direction the point (; 3 3). at the 83. Find the directional derivative of the function w = yz at the point A( ; ; 3) in the direction of s = (4; 3; ). 84. Find the directional derivative of the function w = y z + yz at the point B(; ; ) in direction forming with coordinate aes the angles 6, 45 and 6 respectively. 85. Find the greatest increase of the function z = ln( + y ) at the point C( 3; 4) 6
86. Find the greatest value of the derivative of function given by the equation + y 3 z = at the point (3; ; 4). 87. Find the steepest ascent of the surface z = arctan y at the point (; ). 88. Find the direction of greatest increase of the function f(, y, z) = sin z y cos z at the origin. 89. Find the divergence and curl of the vector field ( F = y ; y z ; z ). 9. Find the divergence and curl of the vector field F = (ln( y ); arctan(z y); yz). 9. Find the divergence and curl of the vector field F = grad w, if w = ln( + y z). 9. Find the divergence and curl of the vector field F = rot G, if G = ( y; y z; z). 93. Find the local etrema of the function z = 4 y + 9y + y and determine their type. 94. Find the local etremum points of the function z = 3 y ( y), satisfying the conditions > and y > and determine their type. 95. Find the local etrema of the function z = + y + y + + y and determine their type. 96. Find the local etrema of the function z = e ( + y ) and determine their type. 97. Find the local etrema of the function z = 3 +y 3 3y and determine their type. 98. Find the greatest and the least value of the function z = + y 4 + 8y in the rectangle bounded by =, y =, = and y =. 99. Find the greatest and the least value of the function z = y in the circle + y 4.. Find the greatest and the least value of the function z = sin + sin y + sin( + y) in the quadrate π, y π. 7
. Find the etremal values of the function z = + under the condition y + y =.. Find the etremal values of the function z = a cos + b cos y under the condition y = π 4. 3. Find the etremal values of the function w = + y + z under the condition + y + z =. 4. The sum of three edges of the rectangular bo, passing one verte is m. Find the dimensions of this rectangular bo so that the volume is the greatest. 5. Find the point on the parabola y = 3 closest to the pointp (; ). 6. Find the point in the plane 3 z = so that the sum of squares of distances from the points A(; ; ) and B(; 3; 4) is the least. 7. Evaluate the double integral ( + y )d, if is the quadrate and y. 8. Evaluate the double integral and 3 y 4. d, if is the quadrate ( + y) 9. Evaluate the double integral d 3 y.. Evaluate the double integral d + (y + y).. Evaluate the double integral d ( + y ).. Sketch the domain of integration and determine the limits of integration for f(; y)d, if is the region bounded by the line y = and the parabola y =. 8
3. Sketch the domain of integration and determine the limits of integration for f(; y)d, if is the parallelogram bounded by the lines y =, y = a, y = and y = a. 4. Sketch the domain of integration and determine the limits of integration for f(; y)d, if is the region bounded by y = + and y =. 5. Sketch the domain of integration and change the order of integration for d f(; y). 3 6. Sketch the domain of integration and change the order of integration + for d f(; y). 7. Sketch the domain of integration and change the order of integration for y f(; y)d. y 8. Sketch the domain of integration and change the order of integration + for d f(; y). 9. Changing the order of integration epress the sum y+ f(; y)d by one double integral. f(; y)d + y. Sketch the domain of integration, determine the limits and evaluate the double integral ( y)d, if is the region given by inequalities and y +. 9
. Sketch the domain of integration, determine the limits and evaluate the double integral ( + y )d, if is bounded by the lines y =, + y = a and =.. Sketch the domain of integration, determine the limits and evaluate the double integral yd, if is the least of segments bounded by the line + y = and circle + y = y. 3. Sketch the domain of integration, determine the limits and evaluate the double integral e +y d, if is the region bounded by y = e, = and y =. 4. Convert the double integral f(; y)d to polar coordinates, if is the region determined by inequalities + y 4 and y. 5. Convert the double integral f(; y)d to polar coordinates, if is bounded by the circles + y = 4 and + y = 8 and the lines y = and y =. 6. Convert the double integral R Ry y f(; y)d to polar coordinates. R 7. Evaluate the double integral by converting it into polar coordinates a a d e +y. 8. Evaluate the double integral by converting it into polar coordinates d 4 y, if is the region determined by inequalities + y 4,, y.
9. Evaluate the double integral by converting it into polar coordinates R R d ln ( + + y ). 3. Evaluate the double integral by converting it into polar coordinates R y d, if is the circle + y R. 3. Evaluate the double integral by converting it into polar coordinates + y d, if is bounded by the part of lemniscate ( + y ) = a ( y ) where. 3. Evaluate the triple integral 33. Evaluate the triple integral a d V y 3 y zdz. ddz, if V is the region ( + y + z + ) 3 bounded by planes =, y =, z = and + y + z =. 34. Evaluate the triple integral yzddz, if V is bounded by the V V surfaces y =, = y, z = y and z =. 35. Convert the triple integral f(; y; z)ddz into cylindrical coordinates, if V is the region bounded by the planes =, y =, z = and cylinders + y = 4 and z = + y. 36. Evaluate the triple integral by converting it into cylindrical coordinates d dz + y +y V 37. Evaluate the triple integral by converting it into cylindrical coordinates z + y ddz, if V is the region determined by the inequalities, z 3 and y.
38. Convert the triple integral V f(; y; z)ddz into spherical coordinates, if V is the region determined by the inequalities + y + z 4, z and y. 39. Evaluate the triple integral by converting it into spherical coordinates R d R R y dz z R R V 4. Evaluate the triple integral by converting it into spherical coordinates + y + z ddz, if V is the region determined by the inequalities y, y and z y. 4. Compute the area bounded by y = 4 and + y = 5. 4. Compute the area bounded byy = 8a3, = y and = provided + 4a a is a positive constant. 43. Compute the volume of solid bounded by the planes z =, y =, y = and = and paraboloid of revolution z = + y. 44. Compute the volume of solid bounded by the hyperbolic paraboloid (saddle surface) z = y and the planes z = and = 3. 45. Compute the volume of solid bounded by the surfaces z = + y, z = ( + y ), y = and y =. 46. Compute the volume of solid bounded by the sphere + y + z = 4 and paraboloid of revolution 3z = + y. 47. Compute the volume of solid determined by the equations y, y, + y 4 and z + y +. 48. Compute the volume of solid determined by the equations +y +z R and + y + z Rz. 49. Compute the line integral ( + y )ds where is the line segment from A(; ) to B(4; 4).
5. Compute the line integral y ds where is the arc of cycloid = a(t sin t), y = a( cos t) between the points O(; ) and C(aπ; ). 5. Compute the line integral ( + y + z)ds where is the arc of heli = a cos t, y = a sin t, z = bt from t = to t = π 5. Compute the line integral yzds where is the quarter of circle = R cos t, y = R sin t, z = R 3, which lies in the first octant. yd + 53. Compute the line integral where is the segment of the + y line y = from (; ) to (; ). 54. Compute the line integral arctan y d where is the arc of parabola y = from O(; ) to A(; ). 55. Compute the line integral ( + y)d + ( y) where AB is the arc AB of ellipse = a cos t, y = b sin t from A(a; ) to B(; b). 56. Compute the line integral yd where is the arc of astroid = a cos 3 t, y = a sin 3 t from t = to t = π. 57. Compute the line integral y d + y = t sin t, y = cos t from t = π 6 to t = π 3. 58. Compute the line integral AB where is the arc of cycloid d + y + zdz + y + z y + z the line segment from A(; ; ) to B(4; 4; 4). where AB is 3
59. Compute the line integral yzd + z + ydz where is the arc of heli = a cos t, y = a sin t, z = bt from t = to t = π. 6. Convert the line integral ( 3 )yd + ( + y 3 ) to the double integral over the region where is positively oriented, smooth, closed curve and the region enclosed by. 6. Convert the line integral e ( cos y)d+e (sin y+y) to the double integral over the region where is positively oriented, smooth, closed curve and the region enclosed by. 6. Use Green s theorem to find ( + y )d + ( + y) where is the contour of triangle ABC with vertices A(; ), B(; ) and C(; ) with positive orientation. 63. Use Green s theorem to find (5 3y)d + ( 4y) where is the circle + y = with positive orientation. 64. Use Green s theorem to find yd + where is the contour of square + y = with positive orientation. 65. Use Green s theorem to find y yd where is the circle + y = 5 with positive orientation. 66. Find the function u, if the total differential is du = d + y. 67. Find the function u, if the total differential is du = (cos y e y )d ( e y + sin y). 68. Evaluate (;) yd +. (;) 4
69. Evaluate (;3) yd +. ( ;) 7. Evaluate (;) d + y + y. (;) 7. Evaluate (+y+z)dσ where S is the part of the plain 4 + y +z = S in the first octant. 7. Evaluate ( + y )dσ where S is the surface cut from the cone S z = + y by the cylinder + y =. 73. Evaluate R y dσ where S is the upper half of the sphere S z = R y. 74. Evaluate + + y dσ where S is the part of the saddle surface S z = y cut by the cylinder + y =. 75. Evaluate dz + yddz + zd where S is that part of the plane S + y + z = which is in the first octant. Choose the side where the normal forms the acute angles with coordinate aes. 76. Evaluate y zd where S is the upper side of the hemisphere S z = R y. 77. Evaluate yzdz where S is the lower side of the hemisphere S z = R y. 78. Evaluate zd + ydz + yzddz+ where S is the inner side S 5
of the pyramid determined by the planes =, y =, z = and + y + z =. 79. Write the general term of the series ( 3 ( ) 5 3 5) + + +.... 8 8. Write the general term of the series 7 + 3 3 4 9 +.... 8. Using the identity k(k + ) = k, find the nth partial sum and k + the sum of the series + 3 + 3 4 +... + k(k + ) +.... 8. Find the nth partial sum and the sum of the series... + 3k(3k + 3) +.... 83. find the nth partial sum and the sum of the series k + + k). 84. Use the Comparison Test to determine whether the series converges or diverges. 85. Use the Comparison Test to determine whether the series converges or diverges. 3 6 + 6 9 + 9 + ( k + k= k= k= k 5 + 3 k (k 3 ) 3 86. Use the d Alembert Test to determine whether the series + 3! + 6 3! + 4! +... + 3 n +... converges or diverges. n! 87. Use the d Alembert Test to determine whether the series k k! converges or diverges. 88. Use the d Alembert Test to determine whether the series converges or diverges. k= k= 3 k (3k + )! 6
89. Use the Cauchy Test to determine whether the series converges or diverges. 9. Use the Cauchy Test to determine whether the series converges or diverges. 9. Use the Cauchy Test to determine whether the series converges or diverges. k= k= arcsin k k 4k + 3 k= ln k k + 3 k + ( ) k k + k k + 9. Use the Integral Test to determine whether the series 4 + 7 + +... + +... converges or diverges. 3k + 93. Use the Integral Test to determine whether the series k(ln k) converges or diverges. k= 94. Use the eibnitz s Test to determine whether the series + 3 + +... converges or diverges. 4 5 6 95. Use the eibnitz s Test to determine whether the series converges or diverges. k= cos kπ k 3 96. oes the series 3 + 5 7 +... + ( )n+ (n ) +... converges conditionally or absolutely? 97. oes the series + 3... + 3 ( )n+ n converges conditionally or absolutely? () +... n 7
98. oes the series ln ln 3 + ln 4 ln 5 +... + ( )n ln n +... converges conditionally or absolutely? In eercises 99. - 3. find the values of for which the functional series is convergent. 99. + + 4 +... + n n +..... + 3 3 4 n +... + ( )n+ 4 n +..... sin + sin 3 + 4 sin 9 +... + n sin 3 n +..... 3. + + + + 3 3 + 3 +... + n n + n +.... ln k (e). k= In eercises 4. - 6. determine whether the functional series can be majorized. 4. + + +... + n n +.... 5. + + + 3 3 +... + n n +.... 6. 7. 8. 9. sin sin sin 3 sin n + + +... + +... π. 3 n In eercises 7. -. find the radius of convergence and the domain of convergence. k= k(k + ). (k) k. k! k= k= k( ) k 3 k. 8
. k ( + 3) k. k! k= In eercises. - 6. epand the function in powers of and determine the domain of convergence.. f() = +.. f() = e. 3. f() = +. 4. f() = sinh. 5. f() = cos. 6. f() = arctan (Remark: integrate the result of the eercise 3. in limits from to ). In eercises 7. -. find the Fourier series epansion of the given π-periodic function defined on a half-open interval. {, π < < 7. f() =, π 8. f() =, if π < π. 9. f() =, if π < π.. f() = sin a, if π < π.. f() = π, if < π. Answers 9. No because the first function is additionally determined, if and sin y.. No because the first function is additionally determined, if < and y <.. ; -; does not eist... 3.. 4. oes not eist. 5.. 6. oes not eist. 7.. 8. 3 y + 4 y ; 3 3 y 4y 4 y. 9. y sin y ; y sin y.. y e y ; y e y.. y cos y 9
y sin y ; cos y+ sin y.. + y ; y 3. ( + y ) ; + y. y 5. y + ln z ; 3y cos ( + y 3 + z 4 ) ; 4z 3 y ( + + y ) + y. y y 4. y ln(+y)+ ; ln(+y)+ + y + y. y + ln z ; z(y + ln z ). 6. cos ( + y 3 + z 4 ) ; cos ( + y 3 + z 4 ). 7. yz yz ; yz ln zy z ; yz ln y z ln y. 8. y + y ( + ) ; y y + y + y ( + ). + y 9. 5 ; 5. 3. ; -. 3. 3 yd. 34. dz = y y. ( 35. dz = cos y cos y + y sin y sin y ) yd + 36. dz = ( + y ) yd yzd. 37. dw = yz + z ln + y ln dz. 38. y tan y 9 9. 39. z = ; dz =. 4. z, 3764; 36 635 6 dz =, 35. 4. 59,84. 4. 53 54. 43. π 6. 44.,6. 45. 49. z + ; 5. dz = 54. ( y ) y( y ). 46. 3. 47. 3. 48.. 4e y z +. 5. + π ; π + π. 5. e ;. sin z (sin d + sin y). 53. + ln ( ln + ln +. 3 4 t ) 3. 56. t + 4. 55. cos ( 3t + t t ) e sin. 57. + u cos + ve + u + v. 58. t( t cos t) sin t sin t sin t t 4. 59. u cos v v sin u cos u u cos v + v cos u ; v cos u u sin v cos v u cos v + v cos u. 6. u ; 4 sinh v cosh v sinh v + cosh v. y( y) 6. + y ( y) ; ( y) + y ( y). 63. y 3 ( y ) y ; 3 y ( y ) y. ( y ) y ; 64. eyz ( + 3yz + y z ). 65. ; 4 ; 3 8. 66. 4 5 ; 3 5 ; 4 5. 67.
. 7. z π. 7. 4 ( 3 ; 9 ; 7 = y + π + = z 3. 7. = y ). 74. + y 4z = π. = y = = z. 73. ( ; ; ); 4 = y + 4 = z + π 4. 3 75. 3 4y 5z = ; = y + 4 3 4 = z 5. 76. z + = ; 5 5 + = y + π = z +. 77. 3 + 4y + 6z = ; = 3 y = z ( 4 6. 79. ; ) (. 8. 4 3 ; 3 ) ( ) 7 ; 4 3 ; 3. 4 ( 8. ; ) ;. 8. 5 3. 83. 3 4 3 3. 84. +. 85.,4; 86. 3 5 4 ; 87.. 88. (; ; ). 89. div F = y + z + ; rot ( y F = z ; z ; ). 9. div F = y y + (z y) + y; rot ( ) F = z + (z y) ; yz; y. y 9. div 3 F = ( + y z) ; rot F = Θ. 9. div F = ; rot F = ( (; ; y z). 93. ocal minimum at 7 ). 94. o- 43 ; 7 43 cal( maimum z ma = 69 at (6; 4). 95. ocal minimum z min = 3 3 3 at 3 3 ; 3 ). 96. There is no local etremum at ( ; ), local 3 minimum at (; ). 97. There is no local etremum at (; ), local minimum at (; ). 98. z min = z(; ) = 3; z ma = z(; ) = 7. 99. z min = z(; ) = z(; ) = 4; z ma = z(; ) = z( ; ) = ( π 4.. z min = z(; ) = ; z ma = z 3 ; π ) = 3 3 3.. z(; ) =.. ( arctan b a ; π 4 arctan b a ( ; ; ); (; ; ); (; ; ). 4. 3, 3 and 3 ). 3. (3; 3; 3); m. 5.
( ) ( 3 8 ; ) 3 ; 6 8 ;. 6. 6 9. 3 3 9.. 4.. 6 35.. 3. 7. a 3 y y+a y f(; y)d. 4. y f(; y)d. 6. d + f(; y) d ( ) 63 ; ;. 7. 8 5. 8. ln 3 6 3 4. y d + f(; y)d + y f(; y). d f(; y). f(; y). 5. y f(; y)d. 8. + y f(; y)d+ 3 f(; y)d 3 y f(; y)d. 9. d y y f(; y).. 7.. 4 3 a4.. 3. e. 4. π dφ f(ρ cos φ; ρ sin φ)ρdρ. 5. arctan dφ 4. 8 cos φ f(ρ cos φ; ρ sin φ)ρdρ 6. π dφ R sin φ π 4 4 cos φ f(ρ cos φ; ρ sin φ)ρdρ. 7. π 4 (ea ). 8. π 6 R sin φ π π. 9. 4 [( + R ) ln( + R ) R ]. 3. 3. 5 a4. 3. 35. π dφ dρ ρ R 3 3 ( π 4 ). 3 a. 33. ln 5 6. 34. 96. f(ρ cos φ; ρ sin φ; z)ρdz. 36. 4 π. 37. 8. 3 38. π dφ π dθ f(r cos φ sin θ; r sin φ sin θ; r cos θ)r sin θdr. 39. π
8 5 πr π R. 4. 8 ; 4. (5 6 ln ). 4. a 6 (π ). 43. 3. 44. 7. 45. 3 35. 46. 9 π 5πR3 π. 47.. 48. 6 6. 49. 4. 5. 56 5 a3. 5. π a + b (a R 4 3 + πb). 5. 3. 53. π ln. 54.. 55. + b 3πab a. 56. 6. π 57. 4 + 3 ln 3. 58. 3 3; 59.. 6. ( 3 + y 3 )d. 6. e yd. 6. 3. 5π 63. 4π. 64.. 65.. 66. u(, y) = 3 (3 + y 3 ) + C. 67. u(, y) = cos y e y + C. 68. 4. 69. 8. 7.. 7. 7. 7. π 3. 73. πr3. 74. 3π. 75.. 76. πr 7 5. 77.. 78. 8. 79. ( k 3k ) k. 8. ( ) k k 6k 5. 8. S n = n + ; S =. 8. S n = 9 9(n + ) ; S = 9. 83. + n + n + ;. 84. Converges. 85. iverges. 86. Converges. 87. Converges. 88. Converges. 89. Converges. 9. Converges. 9. Converges. 9. iverges. 93. Converges. 94. Converges. 95. Converges. 96. Converges absolutely. 97. Converges absolutely. 98. Converges conditionally. 99. < <.... < <.. <. 3. (e ; ). 4. Majorized. 5. Not majorized. 6. Majorized. 7. ; [ ; ]. 8. e ; [ e ; e ). 9. 3; ( ; 5).. ; R.. + 3 +..., converges 3 4 for < <.. +! 3 +..., converges for < 3 3! <. 3. + 4 6 +..., converges for < <. 4. + 3 3! + 5 5! +..., converges for < <. 5. + ( ) n () n, (n)! n= 3
converges for < <. 6. 3 3 + 5 5 7 +..., converges 7 4 sin(k + ) ( ) k+ for. 7.. 8. sin k. π k + k k= k= π 9. 3 + 4 ( ) k sin aπ ( ) k k cos k.. sin k.. k π a k k= k= sin k. k k= 4