CONCEPTS
The following list includes terms that are defined and theorems that are stated in this chapter. For each term or theorem, (1) give a precise definition or statement, (2) state in general terms what it means and (3) describe the types of problems with which it is associated.
Sequence Limit of sequence Squeeze Theorem Infinite series Partial sum Series converges Series diverges Geometric series kth-term test for Harmonic series Integral Test divergence Comparison Test Limit Comparison p-Series
Conditional Test Alternating Series Test
convergence Absolute convergence Alternating harmonic
Ratio Test Root Test series
Radius of Taylor series Power series
convergence Fourier series Taylor polynomial
Taylor’s Theorem Recurrence relation
TRUE OR FALSE
State whether each statement is true or false and briefly explain why. If the statement is false, try to “fix it” by modifying the given statement to a new statement that is true.
1. An increasing sequence diverges to infinity.
2. As n increases, n! increases faster than 10n. 3. If the sequence andiverges, then the series∞
k=1
akdiverges.
4. If andecreases to 0 as n→ ∞, then∞
k=1
akdiverges.
5. If ∞
1 f (x) d x converges, then∞
k=1
akconverges for ak= f (k).
6. If the Comparison Test can be used to determine the conver-gence or diverconver-gence of a series, then the Limit Comparison Test can also determine the convergence or divergence of the series.
7. Using the Alternating Series Test, if lim
k→∞ak = 0, then you can conclude that∞
k=1
akdiverges.
8. The difference between a partial sum of a convergent se-ries and its sum is less than the first neglected term in the series.
9. If a series is conditionally convergent, then the Ratio Test will be inconclusive.
10. A series with all negative terms cannot be conditionally convergent.
11. If∞
k=1|ak| diverges, then∞
k=1
akdiverges.
12. A series may be integrated term-by-term, and the interval of convergence will remain the same.
13. A Taylor series of a function f is simply a power series repre-sentation of f .
14. The more terms in a Taylor polynomial, the better the approximation.
15. The Fourier series of x2converges to x2for all x.
16. A recurrence relation can always be solved to find the solution of a differential equation.
In exercises 1–8, determine whether the sequence converges or diverges. If it converges, give the limit.
1. an = 4
3+ n 2. an = 3n
1+ n 3. an = (−1)n n
n2+ 4 4. an = (−1)n n n+ 4 5. an = 4n
n! 6. an = n!
nn 7. an = cos πn 8. an = cos nπ
n
In exercises 9–18, answer with “converges” or “diverges” or
“can’t tell.”
9. If lim
k→∞ak= 1, then∞
k=1
ak .
10. If lim
k→∞ak= 0, then∞
k=1
ak .
11. If lim
k→∞
ak+1
ak
= 1, then∞
k=1ak .
12. If lim
k→∞
ak+1
ak
= 0, then∞
k=1
ak .
13. If lim
k→∞ak= 1
2, then∞
k=1
ak .
14. If lim
k→∞
ak+1
ak
=1 2, then∞
k=1
ak .
REVIEW EXERCISES
15. If lim
k→∞
k
|ak| =1 2, then∞
k=1
ak .
16. If lim
k→∞k2ak= 0, then∞
k=1ak .
17. If p> 1, then∞
k=1
8
kp .
18. If r> 1, then∞
k=1
ark .
In exercises 19–22, find the sum of the convergent series.
19.
∞ k=0
4 1
2
k
20.
∞ k=1
4 k(k+ 2) 21.
∞ k=0
4−k 22.
∞ k=0
(−1)k 3 4k
In exercises 23 and 24, estimate the sum of the series to within 0.01.
23.
∞ k=0
(−1)k k
k4+ 1 24.
∞ k=0
(−1)k+13 k!
In exercises 25–44, determine if the series converges or diverges.
25.
∞ k=0
2k
k+ 3 26.
∞ k=0
(−1)k 2k k+ 3 27.
∞ k=0
(−1)k 4
√k+ 1 28.
∞ k=0
√ 4 k+ 1 29.
∞ k=1
3k−7/8 30.
∞ k=1
2k−8/7
31.
∞ k=1
√k
k3+ 1 32.
∞ k=1
√ k k3+ 1 33.
∞ k=1
(−1)k4k
k! 34.
∞ k=1
(−1)k2k k
35.
∞ k=1
(−1)kln
1+ 1 k
36.
∞ k=1
cos k√ π k 37.
∞ k=1
2
(k+ 3)2 38.
∞ k=2
4 k ln k 39.
∞ k=1
k!
3k 40.
∞ k=1
k 3k 41.
∞ k=1
e1/k
k2 42.
∞ k=1
1 k√
ln k+ 1
43.
∞ k=1
4k
(k!)2 44.
∞ k=1
k2+ 4 k3+ 3k + 1 In exercises 45–48, determine if the series converges absolutely, converges conditionally or diverges.
45.
∞ k=1
(−1)k k
k2+ 1 46.
∞ k=1
(−1)k 3 k+ 1 47.
∞ k=1
sin k
k3/2 48.
∞ k=1
(−1)k+1 3 ln k+ 1 In exercises 49 and 50, find all values of p for which the series converges.
49.
∞ k=1
2
(3+ k)p 50.
∞ k=1
ekp
In exercises 51 and 52, determine the number of terms necessary to estimate the sum of the series to within 10−6.
51.
∞ k=1
(−1)k 3
k2 52.
∞ k=1
(−1)k2k k!
In exercises 53–56, find a power series representation for the function. Find the radius of convergence.
53. 1
4+ x 54. 2
6− x
55. 3
3+ x2 56. 2
1+ 4x2
In exercises 57 and 58, use the series from exercises 53 and 54 to find a power series and its radius of convergence.
57. ln (4+ x) 58. ln (6− x)
In exercises 59–66, find the interval of convergence.
59.
∞ k=0
(−1)k2xk 60.
∞ k=0
(−1)k(2x)k
61.
∞ k=1
(−1)k2
kxk 62.
∞ k=1
−3√ k
x 2
k
63.
∞ k=0
4
k!(x− 2)k 64.
∞ k=0
k2(x+ 3)k
65.
∞ k=0
3k(x− 2)k 66.
∞ k=0
k 4k(x+ 1)k
In exercises 67 and 68, derive the Taylor series of f (x) about the center x c.
67. f (x)= sin x, c = 0 68. f (x)= 1 x, c = 1
REVIEW EXERCISES
In exercises 69 and 70, find the Taylor polynomial P4(x). Graph f (x) and P4(x).
69. f (x)= ln x, c = 1 70. f (x)= 1
√x, c = 1 In exercises 71 and 72, use the Taylor polynomials from exercises 69 and 70 to estimate the given values. Determine the order of the Taylor polynomial needed to estimate the value to within 10−8.
71. ln 1.2 72. 1
√1.1
In exercises 73 and 74, use a known Taylor series to find a Taylor series of the function and find its radius of convergence.
73. e−3x2 74. sin 4x
In exercises 75 and 76, use the first five terms of a known Taylor series to estimate the value of the integral.
75.
1 0
tan−1x d x 76.
2 0
e−3x2d x
In exercises 77 and 78, derive the Fourier series of the function.
77. f (x)= x, −2 ≤ x ≤ 2 78. f (x)=
0 if−π < x ≤ 0 1 if 0< x ≤ π
In exercises 79–82, graph at least three periods of the function to which the Fourier series expansion of the function converges.
79. f (x)= x2, −1 ≤ x ≤ 1 80. f (x)= 2x, −2 ≤ x ≤ 2 81. f (x)=
−1 if −1 < x ≤ 0 1 if 0< x ≤ 1 82. f (x)=
0 if−2 < x ≤ 0 x if 0< x ≤ 2
83. Suppose you and your friend take turns tossing a coin. The first one to get a head wins. Obviously, the person who goes first has an advantage, but how much of an advantage is it? If you go first, the probability that you win on your first toss is
1
2, the probability that you win on your second toss is 18, the probability that you win on your third toss is321 and so on. Sum a geometric series to find the probability that you win.
84. In a game similar to that of exercise 83, the first one to roll a 4 on a six-sided die wins. Is this game more fair than the pre-vious game? The probabilities of winning on the first, second and third roll are16,21625 and7776625, respectively. Sum a geometric series to find the probability that you win.
In exercises 85 and 86, find the recurrence relation and a general power series solution of the form ∞
n0anxn.
85. y− 2xy− 4y = 0 86. y+ (x − 1)y= 0 In exercises 87 and 88, find the recurrence relation and a general power series solution of the form ∞
n0
an(x− 1)n.
87. y− 2xy− 4y = 0 88. y+ (x − 1)y= 0 In exercises 89 and 90, solve the initial value problem.
89. y− 2xy− 4y = 0, y(0) = 4, y(0)= 2 90. y− 2xy− 4y = 0, y(1) = 2, y(1)= 4
CONNECTIONS
1. The challenge here is to determine∞
k=1
xk
k(k+ 1)as completely as possible. Start by finding the interval of convergence. Find the sum for the special cases (a) x= 0 and (b) x = 1. For 0< x < 1, do the following. (c) Rewrite the series using the partial fractions expansion of 1
k(k+ 1). (d) Because the se-ries converges absolutely, it is legal to rearrange terms. Do so and rewrite the series as x+ x− 1
x
1
2x2+13x3+ 14x4+ · · · . (e) Identify the series in brackets as ∞
k=1
xk
d x, evaluate the series and then integrate term-by-term. (f) Replace the term in brackets in part (d) with its value obtained in part (e).
(g) The next case is for−1 < x < 0. Use the technique in parts (c)–(f) to find the sum. (h) Evaluate the sum at x= −1 using the fact that the alternating harmonic series sums to ln 2. (Used by permission of Virginia Tech Mathematics Contest. Solution suggested by Gregory Minton.)
2. You have used Fourier series to show that∞
k=1
1 k2 = π2
6. Here, you will use a version of Vi`eta’s formula to give an alternative derivation. Start by using a Maclaurin series for sin x to derive a series for f (x)= sin√
√ x
x . Then find the zeros of f (x). Vi`eta’s formula states that the sum of the reciprocals of the zeros of f (x) equals the negative of the coefficient of the linear term in the Maclaurin series of f (x) divided by the constant term. Take this equation and multiply byπ2 to get the desired formula.