Problem 4. Suppose a line segment is divided into two pieces, the longer of length a and the shorter of length b. ,then this ratio is known as the golden ratio or golden mean, denoted 1.618034. (It doesn't a +b If a a +b a matter what a and b are; as long as the ratio will equal .) See the figure below: a a+b (a) Show that is a solution to the equation x2 – x – 1 = 0, and find an exact (non-decimal) expression for e. a + b (Hint: In the equation a let b = 1 and a = x, so b' = x = 4.) b (b) Consider the sequence an defined in the following way: 1 a2 = 1+ 1 1 1 a1 = 1+1, a4 = 1 + 1+ az = 1+ az = 1+ 1 1+ 1+1 1+1 1 1 1+ 1 1+ 1+1 1 1+ 1 1+1 This sequence has a limit (you do not need to prove this), given by 1 L = 1+ 1 1+ 1 1+ 1+ 1+.. What is this limit? (Hint: First explain why L = 1+
Problem 4. Suppose a line segment is divided into two pieces, the longer of length a and the shorter of length b. ,then this ratio is known as the golden ratio or golden mean, denoted 1.618034. (It doesn't a +b If a a +b a matter what a and b are; as long as the ratio will equal .) See the figure below: a a+b (a) Show that is a solution to the equation x2 – x – 1 = 0, and find an exact (non-decimal) expression for e. a + b (Hint: In the equation a let b = 1 and a = x, so b' = x = 4.) b (b) Consider the sequence an defined in the following way: 1 a2 = 1+ 1 1 1 a1 = 1+1, a4 = 1 + 1+ az = 1+ az = 1+ 1 1+ 1+1 1+1 1 1 1+ 1 1+ 1+1 1 1+ 1 1+1 This sequence has a limit (you do not need to prove this), given by 1 L = 1+ 1 1+ 1 1+ 1+ 1+.. What is this limit? (Hint: First explain why L = 1+
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Problem 4. Suppose a line segment is divided into two pieces, the longer of length a and the shorter of length b.
a + b
If
a
then this ratio is known as the golden ratio or golden mean, denoted 1.618034. (It doesn't
b.
=
a
a +b
a
matter what a and b are; as long as
the ratio will equal .) See the figure below:
a
a
a+b
1
(a) Show that is a solution to the equation x2 – x – 1 = 0, and find an exact (non-decimal) expression for 4.
a +b
(Hint: In the equation
a
let b = 1 and a = x, so
b'
= x = 4.)
a
(b) Consider the sequence a, defined in the following way:
1
a2 = 1+
1
1
1
a4 = 1+
1+
az = 1+
1+
a1 = 1+1,
az = 1+
1
1+
1+1
1+1
1
1+
1+1
1
1+
1
1 +
1+1
This sequence has a limit (you do not need to prove this), given by
1
L = 1+
1
1+
1
1 +
1+
1+..
What is this limit? (Hint: First explain why L = 1+
.)
Problem 5. Show that lim Vn! = o0. (Hint: Verify that n! > (n/2)"/2 by observing that half of the factors are
greater than or equal to n/2. You may want to consider the cases where n is even andn is odd separately to verify
that this hint is true.)
Vn!
Problem 6. Let b, =
1
(a) Show that In b, = -
k
In -.
k=1
(b) Show that In b, converges to
In x dx. (Hint: Consider a Riemann sum.)
(c) What is the limit of b,? Justify your answer.
1
Problem 7. Let cn =
1
+ -=
2n
1
n+1
-n + k
k=0
n
(a) Calculate c1, c2, C3, and c4.
1
(Hint: Begin by approximating this integral using two
2n
(b) Show that
dx +
< Cn S
dx +
different Riemann sums of f (x) = x-1 over [n, 2n] with Ar = 1, one Riemann sum using left endpoints, and
the other using right endpoints.)
(c) Find lim cn using the squeeze theorem.
n00
Problem 8. Let a, = H, - In n, where H,, is the nth harmonic number, defined by
n
1
1
+...+ - =
+
3
4
1
Hn = 1+
2
k=1
n+1
dx
(a) Show that an >0 for all n > 1. (Hint: Use a Riemann sum to show that Hn >
(b) Show that an is decreasing. (Hint: Consider an
an+1 to be an area.)
(c) Show that lim an exists (you do not need to find this limit).
n00
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe489eec6-06b4-4f41-8536-03174f230880%2F26b30906-d4dc-4adf-abf4-4b70bef671aa%2Fhrkiurn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4. Suppose a line segment is divided into two pieces, the longer of length a and the shorter of length b.
a + b
If
a
then this ratio is known as the golden ratio or golden mean, denoted 1.618034. (It doesn't
b.
=
a
a +b
a
matter what a and b are; as long as
the ratio will equal .) See the figure below:
a
a
a+b
1
(a) Show that is a solution to the equation x2 – x – 1 = 0, and find an exact (non-decimal) expression for 4.
a +b
(Hint: In the equation
a
let b = 1 and a = x, so
b'
= x = 4.)
a
(b) Consider the sequence a, defined in the following way:
1
a2 = 1+
1
1
1
a4 = 1+
1+
az = 1+
1+
a1 = 1+1,
az = 1+
1
1+
1+1
1+1
1
1+
1+1
1
1+
1
1 +
1+1
This sequence has a limit (you do not need to prove this), given by
1
L = 1+
1
1+
1
1 +
1+
1+..
What is this limit? (Hint: First explain why L = 1+
.)
Problem 5. Show that lim Vn! = o0. (Hint: Verify that n! > (n/2)"/2 by observing that half of the factors are
greater than or equal to n/2. You may want to consider the cases where n is even andn is odd separately to verify
that this hint is true.)
Vn!
Problem 6. Let b, =
1
(a) Show that In b, = -
k
In -.
k=1
(b) Show that In b, converges to
In x dx. (Hint: Consider a Riemann sum.)
(c) What is the limit of b,? Justify your answer.
1
Problem 7. Let cn =
1
+ -=
2n
1
n+1
-n + k
k=0
n
(a) Calculate c1, c2, C3, and c4.
1
(Hint: Begin by approximating this integral using two
2n
(b) Show that
dx +
< Cn S
dx +
different Riemann sums of f (x) = x-1 over [n, 2n] with Ar = 1, one Riemann sum using left endpoints, and
the other using right endpoints.)
(c) Find lim cn using the squeeze theorem.
n00
Problem 8. Let a, = H, - In n, where H,, is the nth harmonic number, defined by
n
1
1
+...+ - =
+
3
4
1
Hn = 1+
2
k=1
n+1
dx
(a) Show that an >0 for all n > 1. (Hint: Use a Riemann sum to show that Hn >
(b) Show that an is decreasing. (Hint: Consider an
an+1 to be an area.)
(c) Show that lim an exists (you do not need to find this limit).
n00
2
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