Series and differential Equation
1. Test the convergence or divergence of the following infinite series, indicating the tests used to arrive at your conclusion:
(a)
(3)
(b)
(4)
(c)
(5)
(Total 12 marks)
2. Express in partial fractions.
Working: 


Answer: ....…………………………………….......... 
(Total 4 marks)
3. Solve the differential equation
= y tan x + 1, 0 £ x < ,
if y = 1 when x = 0.
Working: 


Answer: ....…………………………………….......... 
(Total 4 marks)
4. Discuss the convergence or divergence of the following series:
(a)
(4)
(b) , k is a positive integer.
(5)
(Total 9 marks)
5. (a) Consider the function f (x) = . Show that
f² (x) =
(3)
(b) The third and fourth derivatives of f (x) with respect to x are given by:
f¢² (x) = and f (4)(x) = .
Use Maclaurin’s series to expand f (x), and to show that
. Find the value of k.
(5)
(c) Hence or otherwise show that the value of is
approximately 0.39438.
(4)
(d) Use the substitution x = sin θ to show that the exact value of
is . Hence find an approximation of p to 4 decimal places.
(5)
(Total 17 marks)
6. (a) Find the Maclaurin series of the function g (x) = sin x2 using the series expansion of sin x, ie sin x = .
(1)
(b) Using the Maclaurin series of g (x) = sin x2 evaluate the definite integral
correct to four decimal places.
(5)
(Total 6 marks)
7. (a) Use the ratio test to calculate the radius of convergence of the power
series .
(3)
(b) Using your result from part (a), determine all points x where the power series given in (a) converges.
(5)
(Total 8 marks)
8. Let G = (V, E) be a connected planar graph with v vertices and e edges, in which each cycle has a length of at least c.
(a) Use Euler’s theorem and the fact that the degree of each face is the length of the cycle enclosing it to prove that
e £ (v – 2).
(5)
(b) Find the minimum cycle length in a ΠΊ3,3 graph and use it to prove that the graph is not planar.
(4)
(Total 9 marks)
9. Test whether the following is a convergent series:
If it is, then find an approximation for the sum to two decimal places; if it is not, explain why this is so.
(Total 5 marks)
10. (a) Find Maclaurin’s series expansion for f (x) = ln (1 + x), for 0 ≤ x < 1.
(4)
(b) Rn is the error term in approximating f (x) by taking the sum of the first (n + 1) terms of its Maclaurin’s series. Prove
(2)
(Total 6 marks)
11. Test the convergence or divergence of the following series
(a)
(5)
(b)
(5)
(Total 10 marks)
12. Consider the function , where x Î +.
(a) Show that the derivative
(3)
(b) Sketch the function f (x), showing clearly the local maximum of the function and its horizontal asymptote. You may use the fact that
(5)
(c) Find the Taylor expansion of f (x) about x = e, up to the second degree term, and show that this polynomial has the same maximum value as f (x) itself.
(5)
(Total 13 marks)
13. Determine whether the series converges.
(Total 4 marks)
14. The function y = f (x) satisfies the differential equation
2x2 = x2 + y2 (x > 0)
(a) (i) Using the substitution y = vx, show that
2x = (v – 1)2
(ii) Hence show that the solution of the original differential equation is
y = x – , where c is an arbitrary constant.
(iii) Find the value of c given that y = 2 when x = 1.
(7)
(b) The graph of y = f (x) is shown below. The graph crosses the xaxis at A.
(i) Write down the equation of the vertical asymptote,
(ii) Find the exact value of the xcoordinate of the point A.
(iii) Find the area of the shaded region.
(5)
(Total 12 marks)
15. Consider the differential equation , for x > 0.
(a) Use the substitution y = vx to show that v + x .
(3)
(b) Hence find the solution of the differential equation, given that y = 2 when x = 1.
(4)
(Total 7 marks)
16. Consider the sequence of partial sums {Sn} given by
Sn = n = 1, 2,…
(a) Show that for all positive integers n, S2n ³ Sn + .
(2)
(b) Hence prove that the sequence {Sn} is not convergent.
(5)
(Total 7 marks)
17. For positive integers k and n let
uk = and S2n =
(a) Show that S2n = .
(3)
(b) Hence or otherwise, determine whether the series is convergent or not, justifying your answer.
(4)
(Total 7 marks)
18. Use the substitution y = xv to show that the general solution to the differential equation,
(x2 + y2) + 2xy = 0, x > 0 is
x3 + 3xy2 = k, where k is a constant.
(Total 6 marks)
19. (a) Describe how the integral test is used to show that a series is convergent. Clearly state all the necessary conditions.
(3)
(b) Test the series for convergence.
(5)
(Total 8 marks)
20. (a) Find the first four nonzero terms of the Maclaurin series for
(i) sin x;
(ii)
(4)
(b) Hence find the Maclaurin series for sin x, up to the term containing x5.
(2)
(c) Use the result of part (b) to find .
(2)
(Total 8 marks)
21. Find the Maclaurin series of the function
f (x) = ln (1 + sin x)
up to and including the term in x4.
(Total 8 marks)
22. Find the range of values of x for which the following series is convergent.
(Total 7 marks)
23. (a) Show that the series is convergent.
(3)
Let S = .
(b) Show that for positive integers n ³ 2,
.
(1)
(c) Hence or otherwise show that 1 £ S < 2π.
(4)
(Total 8 marks)
24. Calculate .
(Total 6 marks)
25. Use the integral test to show that the series , is convergent for p > 1.
(Total 6 marks)
26. (a) (i) Find the first four derivatives with respect to x of y = ln (1 + sin x).
(ii) Hence, show that the Maclaurin series, up to the term in x4, for y is
y = x – …
(10)
(b) Deduce the Maclaurin series, up to and including the term in x4, for
(i) y = ln (1 – sin x);
(ii) y = ln cos x;
(iii) y = tan x.
(10)
(c) Hence calculate .
(4)
(Total 24 marks)
27. Consider the differential equation = 1, where êx÷ < 2 and y = 1 when x = 0.
(a) Use Euler’s method with h = 0.25, to find an approximate value of y when x = 1, giving your answer to two decimal places.
(10)
(b) (i) By first finding an integrating factor, solve this differential equation.
Give your answer in the form y = f (x).
(ii) Calculate, correct to two decimal places, the value of y when x = 1.
(10)
(c) Sketch the graph of y = f (x) for 0 £ x £ 1. Use your sketch to explain why your approximate value of y is greater than the true value of y.
(4)
(Total 24 marks)
28. Find the radius of convergence of the series
(Total 5 marks)
29. (a) Given that calculate the value of a, of b and of c.
(5)
(b) (i) Hence, find I =
(ii) If I = when x = 1, calculate the value of the constant of integration giving your answer in the form p + q ln r where p, q, rÎ
(7)
(Total 12 marks)
30. (a) (i) Prove that the alternating series given by converges.
(ii) Approximate the series by finding the 4th partial sum. Give your answer to six decimal places.
(iii) What is the upper bound for the error in this approximation?
(7)
(b) (i) Find the first four nonzero terms of the Maclaurin series for sin x.
(ii) Deduce the nth term of this series.
(iii) Use the ratio test to show that the series is convergent for all values of x.
(iii) Use your series for sin x to find the first four nonzero terms of the Maclaurin series for cos x.
(8)
(Total 15 marks)
31. Given that and y = 1 when x = 0, use Euler’s method with interval h = 05. to find an approximate value of y when x = 1.
(Total 9 marks)
32. (a) Show that = ln sec x + C, where C is a constant.
(2)
(b) Hence find an integrating factor for solving the differential equation
+ y tan x = sec x.
(2)
(c) Solve this differential equation given that y = 2 when x = 0.
Give your answer in the form y = f (x).
(8)
(Total 12 marks)
33. Find the value of
(a) ;
(3)
(b)
(6)
(Total 9 marks)
34. (a) (i) Given that is convergent, where un ³ 0, prove that is also convergent.
(ii) State, with a reason, whether or not the converse of this result is true.
(5)
(b) Use the integral test to determine the set of values of k for which the series
(i) is convergent;
(ii) is divergent.
(10)
(Total 15 marks)
35. Consider the function f defined by
f (x) = arcsin x, for £ 1.
The derivatives of f (x) satisfy the equation
(1 – x2) f (n+2) (x) – (2n + 1) x f (n+1) (x) – n2 f (n) (x) = 0, for n ³ 1.
The coefficient of xn in the Maclaurin series for f (x) is denoted by an. You may assume that the series contains only odd powers of x.
(a) (i) Show that, for n ³ 1, (n + 1) (n + 2) an+2 = n2an.
(ii) Given that a1 = 1, find an expression for an in terms of n, valid for odd n ³ 3.
(7)
(b) Find the radius of convergence of this Maclaurin series.
(4)
(c) Find an approximate value for p by putting x = and summing the first three nonzero terms of this series. Give your answer to four significant figures.
(4)
(Total 15 marks)
36. The general term of a sequence is given by the formula an = nÎ +.
(a) Given that = L, where LÎ , find the value of L.
(3)
(b) Find the smallest value of NÎ + such that < 10–3 for all n ³ N.
(6)
(Total 9 marks)
37. Consider the differential equation = where x, y > 0.
(a) Show that the differential equation is homogeneous.
(2)
(b) Find the general solution of the differential equation, giving your answer in the form y2 = f (x).
(7)
(c) Solve the differential equation, given that y = 2 when x = 1.
(3)
(Total 12 marks)
38. Estimate the range of values of x for which the Maclaurin approximation
is accurate to within 0.005.
(Total 5 marks)
39. Determine whether the series is convergent or divergent.
(Total 7 marks)
40. Consider the differential equation with boundary condition y = 1 when x = 0.
Use four steps of Euler’s method starting at x = 0, with interval h = 0.1, to find an approximate value for y when x = 0.4.
(Total 10 marks)
41. Consider the series S = .
(a) Use the ratio test to prove that this series is convergent.
(4)
(b) Use a comparison test to show that S < 2.
(4)
(c) Write down the exact value of S.
(1)
(Total 9 marks)
42. (a) Show that the polynomial approximation for ln x in the interval [0.5, 1.5] obtained by taking the first three nonzero terms of the Taylor series about x =1 is given by
ln x » .
(7)
(b) Given = x ln x – x + C, show by integrating the above series that another approximation to ln x is given by
ln x »
(6)
(c) Which is the better approximation when x = 1.5?
(4)
(Total 17 marks)
43. (a) Show that
(3)
(b) Find the solution to the homogeneous differential equation
x2 = x2 + xy – y2, given that y = when x = 1
Give your answer in the form y = g (x).
(16)
(Total 19 marks)
44. (a) (i) Find In = where a is a positive constant and n is a positive integer.
(ii) Determine
(6)
(b) Using l’Hôpital’s Rule find
where b is a nonzero constant different from ±1.
(9)
(Total 15 marks)
45. (a) Use l’Hôpital’s Rule to find
(i)
(ii)
(8)
(b) Giving a reason, state whether the following argument is correct or incorrect.
“Using l’Hôpital’s Rule, ”
(2)
(Total 10 marks)
46. Given that the Maclaurin series for esin x is a + bx + cx2 + dx3 + ..., find the values of a, b, c and d.
(Total 8 marks)
47. Consider the infinite series .
(a) Show that the series is convergent.
(3)
(b) (i) Express in partial fractions.
(ii) Hence find .
(9)
(Total 12 marks)
48. (a) Use integration by parts to show that
(4)
Consider the differential equation – y cos x = sin x cos x.
(b) Find an integrating factor.
(3)
(c) Solve the differential equation, given that y = − 2 when x = 0. Give your answer in the form y = f (x).
(9)
(Total 16 marks)
49. Find the interval of convergence of the series
(Total 14 marks)
50. Find
(a)
(4)
(b)
(5)
(Total 9 marks)
51. (a) Sketch on graph paper the slope field for the differential equation = x – y at the points (x, y) where xÎ {0, 1, 2, 3, 4} and yÎ {0, 1, 2, 3, 4}. Use a scale of 2 cm for 1 unit on both axes.
(3)
(b) On the slope field sketch the curve that passes through the point (0, 3).
(1)
(c) Solve the differential equation to find the equation of this curve.
Give your answer in the form y = f (x).
(10)
(Total 14 marks)
52. (a) Find constants A and B such that
(4)
(b) Find an expression for Sn, the nth partial sum of
(4)
(c) Hence show that the series converges.
(2)
(Total 10 marks)
53. Find the values of p for which dx converges.
(Total 7 marks)
54. Consider the differential equation + (2x – 1)y = 0 given that y = 2 when x = 0.
(a) (i) Show that = (1 – 2x) –
(ii) By finding the values of successive derivatives at x = 0 obtain a Maclaurin series for y up to and including the x4 term.
(10)
(b) A local maximum value of y occurs when x = 0.5. Use your series to calculate an approximation to this maximum value.
(2)
(c) Use Euler’s method with a step value of 0.1 to obtain a second approximation for the maximum value of y. Set out your solution in tabular form.
(6)
(d) How can each of the approximations found in (b) and (c) be made more accurate?
(2)
(Total 20 marks)
55. Calculate the following limits
(a)
(3)
(b)
(5)
(Total 8 marks)
56. Solve the differential equation
given that y =1 when x =
(Total 9 marks)
57. (a) The function f is defined by f (x) =
(i) Obtain an expression for f (n) (x), the nth derivative of f (x) with respect to x.
(ii) Hence derive the Maclaurin series for f (x) up to and including the term in x4.
(iii) Use your result to find a rational approximation to f
(iv) Use the Lagrange error term to determine an upper bound to the error in this approximation.
(13)
(b) Use the integral test to determine whether the series is convergent or divergent.
(9)
(Total 22 marks)
58. (a) (i) State the domain and range of the function f (x) = arcsin (x).
(ii) Determine the first two nonzero terms in the Maclaurin series for f (x).
(8)
(b) Use the small angle approximation
cos (y) » 1
to find a series for cos (arcsin (x)) up to and including the term in x4.
(4)
(c) (i) Find the Maclaurin series for (p + qx2)r up to and including the term in x4 where p, q, rÎ .
(ii) Find values of p, q, r such that your series in (c) (i) is identical to your answer to (b). Comment on this result.
(7)
(Total 19 marks)
59. (a) Determine , where a is a positive constant, not equal to 1.
(3)
(b) Calculate
(3)
(c) Show that = 1.
(4)
(Total 10 marks)
60. (a) A homogeneous differential equation has the form
Show that the substitution v = leads to a differential equation which can be solved by separation of variables.
(3)
(b) Show that the linear change of variables X = x −1, Y = y − 2, transforms the equation
to a homogeneous form.
Hence solve this equation.
(11)
(Total 14 marks)
61. (a) (i) By considering the graph of y = between x = a and x = a +1, show that
, for all a > 0.
(ii) Deduce that
(iii) Using (i) find p, qÎ such that p < ln 3 < q.
(8)
(b) Given that Hn is the nth partial sum of the harmonic series, show that
Hn –1 < ln n < Hn for n > 1.
(5)
(c) If γn = Hn − ln n, prove that the terms of the sequence {γn : n ³1} decrease as n increases.
(4)
(Total 17 marks)
62. Determine whether the series is convergent or divergent.
(Total 6 marks)
63. (a) Using l’Hopital’s Rule, show that = 0.
(2)
(b) Determine
(5)
(c) Show that the integral is convergent and find its value.
(2)
(Total 9 marks)
64. Consider the differential equation
(a) Find an integrating factor for this differential equation.
(5)
(b) Solve the differential equation given that y =1 when x =1, giving your answer in the form y = f (x).
(8)
(Total 13 marks)
65.
The diagram shows part of the graph of y = together with line segments parallel to the coordinate axes.
(a) Using the diagram, show that
(3)
(b) Hence find upper and lower bounds for
(12)
(Total 15 marks)
66. The function f is defined by
f (x) =
(a) Write down the value of the constant term in the Maclaurin series for f (x).
(1)
(b) Find the first three derivatives of f (x) and hence show that the Maclaurin series for f (x) up to and including the x3 term is
(6)
(c) Use this series to find an approximate value for ln 2.
(3)
(d) Use the Lagrange form of the remainder to find an upper bound for the error in this approximation.
(5)
(e) How good is this upper bound as an estimate for the actual error?
(2)
(Total 17 marks)
67. (a) Find the value of
(3)
(b) By using the series expansions for and cos x evaluate
(7)
(Total 10 marks)
68. Find the exact value of
(Total 9 marks)
69. A curve that passes through the point (1, 2) is defined by the differential equation
(a) (i) Use Euler’s method to get an approximate value of y when x = 1.3, taking steps of 0.1. Show intermediate steps to four decimal places in a table.
(ii) How can a more accurate answer be obtained using Euler’s method?
(5)
(b) Solve the differential equation giving your answer in the form y = f (x).
(9)
(Total 14 marks)
70. (a) Given that y = ln cos x, show that the first two nonzero terms of the Maclaurin series for y are
(8)
(b) Use this series to find an approximation in terms of p for ln 2.
(6)
(Total 14 marks)
71. (a) Find the radius of convergence of the series
(6)
(b) Determine whether the series is convergent or divergent.
(7)
(Total 13 marks)
72. Solve the following differential equation
(x + 1)(x + 2) = x + 1
giving your answer in the form y = f (x).
(Total 11 marks)
73. The function f is defined by f (x) =1n (1 + sin x).
(a) Show that f ′′ (x) =
(4)
(b) Determine the Maclaurin series for f (x) as far as the term in x4.
(6)
(c) Deduce the Maclaurin series for ln (1− sin x) as far as the term in x4.
(2)
(d) By combining your two series, show that ln sec x =
(4)
(e) Hence, or otherwise, find
(2)
(Total 18 marks)
74. Let Sn =
(a) Show that, for n ³ 2, S2n > Sn +
(3)
(b) Deduce that > S2 +
(7)
(c) Hence show that the sequence is divergent.
(3)
(Total 13 marks)