Solved: The hailstone sequence Here is a fascinating

Give an example of a nonincreasing sequence with a limit.

Give an example of a nondecreasing sequence without a limit.

Give an example of a bounded sequence that has a limit.

Give an example of a bounded sequence without a limit.

For what values of r does the sequence 5rn6 converge? Diverge?

Explain how the methods used to find the limit of a function as xS _ are used to find the limit of a sequence.

Compare the growth rates of 5n1006 and 5en>1006 as nS _.

Explain how two sequences that differ only in their first ten terms can have the same limit.

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e n3 n4 + 1 f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e n12 3n12 + 4 f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e 3n3 - 1 2n3 + 1 f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e 2en + 1 en f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e 3n + 1 + 3 3n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e k 29k2 + 1 f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 5tan-1 n6

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 52n2 + 1 - n6

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e tan-1 n n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 5n2>n6

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e a1 + 2 n b n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e a n n + 5 b n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e B a1 + 1 2n b n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e a1 + 4 n b 3n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e n en + 3n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e ln 11>n2 n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e a 1 n b 1>n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e a1 - 4 n b n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 5bn 6, where bn = e n>1n + 12 if n 5000 ne-n if n 7 5000

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 5ln 1n3 + 12 - ln 13n3 + 10n26

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 5ln sin 11>n2 + ln n6

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 5n11 - cos11>n226

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e n sin 6 n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e 1-12n n f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e 1-12n n n + 1 f

934. Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. e 1-12n + 1 n2 2n3 + n f

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = sin np 2

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = 1-12n n n + 1

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = sin 1np>32 1n

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = 3n 3n + 4n

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = 1 + cos 1 n

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = e-n 2 sin1e-n2

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = e-n cos n

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = ln n n1.1

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = 1-12n 2n n

3544. Limits of sequences and graphing Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an = cot a np 2n + 2 b

4552. Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 50.2n6

4552. Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 51.2n6

4552. Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 51-0.72n6

4552. Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 551-1.012n6

4552. Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 51.00001n6

4552. Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 52n + 13-n6

4552. Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 51-2.52n6

4552. Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 51001-0.0032n6

5358. Squeeze Theorem Find the limit of the following sequences or state that they diverge. e cos n n f

5358. Squeeze Theorem Find the limit of the following sequences or state that they diverge. e sin 6n 5n f

5358. Squeeze Theorem Find the limit of the following sequences or state that they diverge. e sin n 2n f

5358. Squeeze Theorem Find the limit of the following sequences or state that they diverge. e cos 1np>22 1n f

5358. Squeeze Theorem Find the limit of the following sequences or state that they diverge. e 2 tan-1 n n3 + 4 f

5358. Squeeze Theorem Find the limit of the following sequences or state that they diverge. e n sin3 1np>22 n + 1 f

Periodic dosing Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume also that aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated. a. Find a recurrence relation for the sequence 5dn6 that gives the amount of drug in the blood after the nth dose, where d1 = 80. b. Using a calculator, determine the limit of the sequence. In the long run, how much drug is in the persons blood? c. Confirm the result of part (b) by finding the limit of 5dn6 directly.

A car loan Marie takes out a $20,000 loan for a new car. The loan has an annual interest rate of 6% or, equivalently, a monthly interest rate of 0.5%. Each month, the bank adds interest to the loan balance (the interest is always 0.5% of the current balance), and then Marie makes a $200 payment to reduce the loan balance. Let Bn be the loan balance immediately after the nth payment, where B0 = $20,000. a. Write the first five terms of the sequence 5Bn6. b. Find a recurrence relation that generates the sequence 5Bn6. c. Determine how many months are needed to reduce the loan balance to zero.

A savings plan James begins a savings plan in which he deposits $100 at the beginning of each month into an account that earns 9% interest annually or, equivalently, 0.75% per month. To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits $100. Let Bn be the balance in the account after the nth deposit, where B0 = $0. a. Write the first five terms of the sequence 5Bn6. b. Find a recurrence relation that generates the sequence 5Bn6. c. How many months are needed to reach a balance of $5000?

Diluting a solution A tank is filled with 100 L of a 40% alcohol solution (by volume). You repeatedly perform the following operation: Remove 2 L of the solution from the tank and replace them with 2 L of 10% alcohol solution. a. Let Cn be the concentration of the solution in the tank after the nth replacement, where C0 = 40%. Write the first five terms of the sequence 5Cn6. b. After how many replacements does the alcohol concentration reach 15%? c. Determine the limiting (steady-state) concentration of the solution that is approached after many replacements.

6368. Growth rates of sequences Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. e n! nn f

6368. Growth rates of sequences Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. e 3n n! f

6368. Growth rates of sequences Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. e n10 ln20 n f

6368. Growth rates of sequences Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. e n10 ln1000 n f

6368. Growth rates of sequences Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. e n1000 2n f

6368. Growth rates of sequences Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. e en>10 2n f

6974. Formal proofs of limits Use the formal definition of the limit of a sequence to prove the following limits. lim nS_ 1 n = 0

6974. Formal proofs of limits Use the formal definition of the limit of a sequence to prove the following limits. lim nS_ 1 n2 = 0

6974. Formal proofs of limits Use the formal definition of the limit of a sequence to prove the following limits. lim nS_ 3n2 4n2 + 1 = 3 4

6974. Formal proofs of limits Use the formal definition of the limit of a sequence to prove the following limits. lim nS_ b-n = 0, for b 7 1

6974. Formal proofs of limits Use the formal definition of the limit of a sequence to prove the following limits. lim nS_ cn bn + 1 = c b , for real numbers b 7 0 and c 7 0

6974. Formal proofs of limits Use the formal definition of the limit of a sequence to prove the following limits. lim nS_ n n2 + 1 = 0

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If lim nS_ an = 1 and lim nS_ bn = 3, then lim nS_ bn an = 3. b. If lim nS_ an = 0 and lim nS_ bn = _, then lim nS_ an bn = 0. c. The convergent sequences 5an6 and 5bn6 differ in their first 100 terms, but an = bn, for n 7 100. It follows that lim nS_ an = lim nS_ bn. d. If 5an6 = 51, 12 , 13 , 14 , 15 , c6 and 5bn6 = 51, 0, 12 , 0, 13 , 0, 14 , 0, c6, then lim nS_ an = lim nS_ bn. e. If the sequence 5an6 converges, then the sequence 51-12n an6 converges. f. If the sequence 5an6 diverges, then the sequence 50.000001 an6 diverges.

7677. Reindexing Express each sequence 5an6 n = 1 as an equivalent sequence of the form 5bn6 n = 3. 52n + 16_ n = 1

7677. Reindexing Express each sequence 5an6 n = 1 as an equivalent sequence of the form 5bn6 n = 3. 5n2 + 6n - 96_ n = 1

7885. More sequences Evaluate the limit of the following sequences or state that the limit does not exist. an = L n 1 x -2 dx

7885. More sequences Evaluate the limit of the following sequences or state that the limit does not exist. an = 75n - 1 99n + 5n sin n 8n

7885. More sequences Evaluate the limit of the following sequences or state that the limit does not exist. an = tan-1 a 10n 10n + 4 b

7885. More sequences Evaluate the limit of the following sequences or state that the limit does not exist. an = cos 10.99n2 + 7n + 9n 63n

7885. More sequences Evaluate the limit of the following sequences or state that the limit does not exist. an = 4n + 5n! n! + 2n

7885. More sequences Evaluate the limit of the following sequences or state that the limit does not exist. an = 6n + 3n 6n + n100

7885. More sequences Evaluate the limit of the following sequences or state that the limit does not exist. an = n8 + n7 n7 + n8 ln n

7885. More sequences Evaluate the limit of the following sequences or state that the limit does not exist. an = 7n n7 5n

8690. Sequences by recurrence relations Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and>or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. an + 1 = 12 an + 2; a0 = 5

8690. Sequences by recurrence relations Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and>or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. an + 1 = 2an 11 - an2; a0 = 0.3

8690. Sequences by recurrence relations Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and>or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. an + 1 = 12 1an + 2>an2; a0 = 2

8690. Sequences by recurrence relations Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and>or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. an + 1 = 4an 11 - an2; a0 = 0.5

8690. Sequences by recurrence relations Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and>or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. an + 1 = 12 + an; a0 = 1

Crossover point The sequence 5n!6 ultimately grows faster than the sequence 5bn6, for any b 7 1, as nS _. However, bn is generally greater than n! for small values of n. Use a calculator to determine the smallest value of n such that n! 7 bn for each of the cases b = 2, b = e, and b = 10.

Fish harvesting A fishery manager knows that her fish population naturally increases at a rate of 1.5% per month, while 80 fish are harvested each month. Let Fn be the fish population after the nth month, where F0 = 4000 fish. a. Write out the first five terms of the sequence 5Fn6. b. Find a recurrence relation that generates the sequence 5Fn6. c. Does the fish population decrease or increase in the long run? d. Determine whether the fish population decreases or increases in the long run if the initial population is 5500 fish. e. Determine the initial fish population F0 below which the population decreases.

The hungry heifer A heifer weighing 200 lb today gains 5 lb per day with a food cost of 45>day. The price for heifers is 65>lb today but is falling 1>day. a. Let hn be the profit in selling the heifer on the nth day, where h0 = 1200 lb2 # 1$0.65>lb2 = $130. Write out the first 10 terms of the sequence 5hn6. b. How many days after today should the heifer be sold to maximize the profit?

Sleep model After many nights of observation, you notice that if you oversleep one night, you tend to undersleep the following night, and vice versa. This pattern of compensation is described by the relationship xn + 1 = 1 2 1xn + xn - 12, for n = 1, 2, 3, c, where xn is the number of hours of sleep you get on the nth night and x0 = 7 and x1 = 6 are the number of hours of sleep on the first two nights, respectively. a. Write out the first six terms of the sequence 5xn6 and confirm that the terms alternately increase and decrease. b. Show that the explicit formula xn = 19 3 + 2 3 a - 1 2 b n , for n 0, generates the terms of the sequence in part (a). c. What is the limit of the sequence?

Calculator algorithm The CORDIC (COordinate Rotation DIgital Calculation) algorithm is used by most calculators to evaluate trigonometric and logarithmic functions. An important number in the CORDIC algorithm, called the aggregate constant, is given by the infinite productq _ n = 0 2n 21 + 22n , where q N n = 0 an represents the product a0 # a1 gaN. This infinite product is the limit of the sequence e q 0 n = 0 2n 21 + 22n , q 1 n = 0 2n 21 + 22n , q 2 n = 0 2n 21 + 22n , cf. Estimate the value of the aggregate constant. (See the Guided Project CORDIC algorithms: How your calculator works.)

Bounded monotonic proof Use mathematical induction to prove that the drug dose sequence in Example 5, dn + 1 = 0.5dn + 100, d1 = 100, for n = 1, 2, 3, c, is bounded and monotonic.

Repeated square roots Consider the expression 41 + 31 + 21 + 11 + g, where the process continues indefinitely. a. Show that this expression can be built in steps using the recurrence relation a0 = 1, an + 1 = 11 + an, for n = 0, 1, 2, 3, c. Explain why the value of the expression can be interpreted as lim nS_ an, provided the limit exists. b. Evaluate the first five terms of the sequence 5an6. c. Estimate the limit of the sequence. Compare your estimate with 11 + 152>2, a number known as the golden mean. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. e. Repeat the preceding analysis for the expression 4p + 3p + 2p + 1p + g, where p 7 0. Make a table showing the approximate value of this expression for various values of p. Does the expression seem to have a limit for all positive values of p?

A sequence of products Find the limit of the sequence 5an 6_ n = 2 = e a1 - 1 2 b a1 - 1 3 bga1 - 1 n b f _ n = 2 .

Continued fractions The expression 1 + 1 1 + 1 1 + 1 1 + 1 1 + f where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation a0 = 1, an + 1 = 1 + 1>an , for n = 0, 1, 2, 3, c. Explain why the value of the expression can be interpreted as lim nS_ an, provided the limit exists. b. Evaluate the first five terms of the sequence 5an6. c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with 11 + 152>2, a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of a + b a + b a + b a + b a + f , where a and b are positive real numbers.

Tower of powers For a positive real number p, the tower of exponents p p p a continues indefinitely and the expression is ambiguous. The tower could be built from the top as the limit of the sequence 5p p, 1p p2p, 11p p2p2p, c6, in which case the sequence is defined recursively as an + 1 = an p 1building from the top2, (1) where a1 = p p. The tower could also be built from the bottom as the limit of the sequence 5p p, p1p p2, p1p1p p22, c6, in which case the sequence is defined recursively as an + 1 = pan 1building from the bottom2, (2) where again a1 = p p. a. Estimate the value of the tower with p = 0.5 by building from the top. That is, use tables to estimate the limit of the sequence defined recursively by (1) with p = 0.5. Estimate the maximum value of p 7 0 for which the sequence has a limit. b. Estimate the value of the tower with p = 1.2 by building from the bottom. That is, use tables to estimate the limit of the sequence defined recursively by (2) with p = 1.2. Estimate the maximum value of p 7 1 for which the sequence has a limit.

Fibonacci sequence The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about a.d. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation fn + 1 = fn + fn - 1, for n = 1, 2, 3, c, where f0 = 1, f1 = 1. Each term of the sequence is the sum of its two predecessors. a. Write out the first ten terms of the sequence. b. Is the sequence bounded? c. Estimate or determine w = lim nS_ fn + 1 fn , the ratio of the successive terms of the sequence. Provide evidence that w = 11 + 152>2, a number known as the golden mean. d. Use induction to verify the remarkable result that fn = 1 15 1wn - 1-12nw-n2.

Arithmetic-geometric mean Pick two positive numbers a0 and b0 with a0 7 b0, and write out the first few terms of the two sequences 5an6 and 5bn6: an + 1 = an + bn 2 , bn + 1 = 1an bn , for n = 0, 1, 2c. (Recall that the arithmetic mean A = 1p + q2>2 and the geometric mean G = 1pq of two positive numbers p and q satisfy A G.) a. Show that an 7 bn for all n. b. Show that 5an6 is a decreasing sequence and 5bn6 is an increasing sequence. c. Conclude that 5an6 and 5bn6 converge. d. Show that an + 1 - bn + 1 6 1an - bn2>2 and conclude that lim nS_ an = lim nS_ bn. The common value of these limits is called the arithmetic-geometric mean of a0 and b0, denoted AGM1a0, b02. e. Estimate AGM112, 202. Estimate Gauss constant 1>AGM11, 122.

The hailstone sequence Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer N and call it a0. This is the seed of a sequence. The rest of the sequence is generated as follows: For n = 0, 1, 2, c an + 1 = e an>2 if an is even 3an + 1 if an is odd. However, if an = 1 for any n, then the sequence terminates. a. Compute the sequence that results from the seeds N = 2, 3, 4, c, 10. You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers N, the sequence terminates after a finite number of terms. b. Now define the hailstone sequence 5Hk6, which is the number of terms needed for the sequence 5an6 to terminate starting with a seed of k. Verify that H2 = 1, H3 = 7, and H4 = 2. c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?

Prove that if 5an6 V 5bn6 (as used in Theorem 8.6), then 5can6 V 5dbn6, where c and d are positive real numbers.

Convergence proof Consider the sequence defined by an + 1 = 23an, a1 = 13, for n 1. a. Show that 5an6 is increasing. b. Show that 5an6 is bounded between 0 and 3. c. Explain why lim nS_ an exists. d. Find lim nS_ an.

106110. Comparing sequences In the following exercises, two sequences are given, one of which initially has smaller values, but eventually overtakes the other sequence. Find the sequence with the larger growth rate and the value of n at which it overtakes the other sequence. an = 2n and bn = 2 ln n, n 3

106110. Comparing sequences In the following exercises, two sequences are given, one of which initially has smaller values, but eventually overtakes the other sequence. Find the sequence with the larger growth rate and the value of n at which it overtakes the other sequence. an = en>2 and bn = n5, n 2

106110. Comparing sequences In the following exercises, two sequences are given, one of which initially has smaller values, but eventually overtakes the other sequence. Find the sequence with the larger growth rate and the value of n at which it overtakes the other sequence. an = n1.001 and bn = ln n10, n 1

106110. Comparing sequences In the following exercises, two sequences are given, one of which initially has smaller values, but eventually overtakes the other sequence. Find the sequence with the larger growth rate and the value of n at which it overtakes the other sequence. an = n! and bn = n0.7n, n 2

106110. Comparing sequences In the following exercises, two sequences are given, one of which initially has smaller values, but eventually overtakes the other sequence. Find the sequence with the larger growth rate and the value of n at which it overtakes the other sequence. an = n10 and bn = n9 ln3 n, n 7

Comparing sequences with a parameter For what values of a does the sequence 5n!6 grow faster than the sequence 5nan6? (Hint: Stirlings formula is useful: n! _ 22pn nne-n, for large