a. Calculate the first 5 terms of the sequence an = (1+1)", expressing your answers as fractions and as decimals to 3dp. b. Prove that the sequence is increasing. Actually this is a bit hard, but proceed as follows.
a. Calculate the first 5 terms of the sequence an = (1+1)", expressing your answers as fractions and as decimals to 3dp. b. Prove that the sequence is increasing. Actually this is a bit hard, but proceed as follows.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
100%
![a. Calculate the first 5 terms of the sequence an
as fractions and as decimals to 3dp.
b. Prove that the sequence is increasing. Actually this is a bit hard, but proceed as
follows.
Note that an =
Deduce that
n+1
= (₁ + ² ) *** / (₁+²). . Hence, use algebra to show that
an+1
an
• Deduce that
Bernoulli's inequality states that for any x ≥-1, (1+x)" ≥ 1+nx. Use this to
deduce that
(1+x)"=1+n+
1
Show that <
k!
and simplify the last expression.
c. Now prove that an is bounded above. This, too, is subtle.
The Binomial Theorem tells us that
n(n-1)
2!
1
(₁+²)"
1+
1
2k
n
- (₁ + ²) "₁
n+1
1
=
- (1 - (n +² 1²) "* ¹ (1+²)
an+1
1
dan 1 ≥ (1 - 2 + ₁) (₁ + ² )
>
n+1
+...
, expressing your answers
+
n(n-1)(n-2)
3!
-23
n(n − 1) ... ((n − (n − 1))n-
(n − 1)!
1 1
< 1+1+ + + +
2! 3!
for any integer k ≥ 4.
(₁+1) "<
+ £ 12/10
2k
k=0
and that the partial sum is <2. Thus an < 3 for all n.
+...
+
1
1
+
(n 1)! n!
n!
21 22
The Monotone Convergence Theorem tells us that the sequence converges to a limit
<3. In fact, that limit is e.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01b4eff1-1c92-4d86-9563-264f29556131%2F64b9b77e-eb3d-46a7-967b-514887223513%2Fcbs1m5f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a. Calculate the first 5 terms of the sequence an
as fractions and as decimals to 3dp.
b. Prove that the sequence is increasing. Actually this is a bit hard, but proceed as
follows.
Note that an =
Deduce that
n+1
= (₁ + ² ) *** / (₁+²). . Hence, use algebra to show that
an+1
an
• Deduce that
Bernoulli's inequality states that for any x ≥-1, (1+x)" ≥ 1+nx. Use this to
deduce that
(1+x)"=1+n+
1
Show that <
k!
and simplify the last expression.
c. Now prove that an is bounded above. This, too, is subtle.
The Binomial Theorem tells us that
n(n-1)
2!
1
(₁+²)"
1+
1
2k
n
- (₁ + ²) "₁
n+1
1
=
- (1 - (n +² 1²) "* ¹ (1+²)
an+1
1
dan 1 ≥ (1 - 2 + ₁) (₁ + ² )
>
n+1
+...
, expressing your answers
+
n(n-1)(n-2)
3!
-23
n(n − 1) ... ((n − (n − 1))n-
(n − 1)!
1 1
< 1+1+ + + +
2! 3!
for any integer k ≥ 4.
(₁+1) "<
+ £ 12/10
2k
k=0
and that the partial sum is <2. Thus an < 3 for all n.
+...
+
1
1
+
(n 1)! n!
n!
21 22
The Monotone Convergence Theorem tells us that the sequence converges to a limit
<3. In fact, that limit is e.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781319050740/9781319050740_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
![Precalculus](https://www.bartleby.com/isbn_cover_images/9780135189405/9780135189405_smallCoverImage.gif)
![Calculus: Early Transcendental Functions](https://www.bartleby.com/isbn_cover_images/9781337552516/9781337552516_smallCoverImage.gif)
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning