8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
question8 induction
![### Mathematical Proofs and Concepts
1. **Prove by contradiction that \(6n + 5\) is odd for all integers \(n\).**
2. **Prove that for all integers \(n\), if \(3n + 5\) is even, then \(n\) is odd.**
*(Hint: Prove the contrapositive.)*
3. **Prove the inequality:**
\[
|x + y| \leq |x| + |y|
\]
for all real numbers \(x\) and \(y\).
4. **Prove that there does not exist a smallest positive real number.**
*(In other words, prove that there does not exist a positive real number \(x\) such that \(x < y\) for all positive real numbers \(y\).)*
5. **Prove that if \(x\) is rational and \(y\) is irrational, then \(x + y\) is irrational.**
*Recall that an irrational number is a real number which is not rational. You may use the fact that the rational numbers are closed under addition - if \(a\) and \(b\) are rational numbers, then \(a + b\) is rational as well.*
6. **Prove that if \(a\), \(b\), and \(c\) are positive real numbers with \(ab = c\), then \(a \leq \sqrt{c}\) or \(b \leq \sqrt{c}\).**
7. **Prove that**
\[
\sum_{i=1}^{n} i^3 = \frac{1}{4}n^2(n+1)^2
\]
for all positive integers \(n\).
8. **Prove that**
\[
\sum_{i=1}^{n} i \cdot i! = (n+1)! - 1
\]
for all positive integers \(n\).
9. **Prove that**
\[
2^n < n!
\]
for all positive integers \(n\) such that \(n \geq 4\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67a79aa2-f715-406c-8bee-8178252bb86d%2F7e9a64df-f270-4c35-a8a7-e19fa26aabb5%2Fbj48aahg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Proofs and Concepts
1. **Prove by contradiction that \(6n + 5\) is odd for all integers \(n\).**
2. **Prove that for all integers \(n\), if \(3n + 5\) is even, then \(n\) is odd.**
*(Hint: Prove the contrapositive.)*
3. **Prove the inequality:**
\[
|x + y| \leq |x| + |y|
\]
for all real numbers \(x\) and \(y\).
4. **Prove that there does not exist a smallest positive real number.**
*(In other words, prove that there does not exist a positive real number \(x\) such that \(x < y\) for all positive real numbers \(y\).)*
5. **Prove that if \(x\) is rational and \(y\) is irrational, then \(x + y\) is irrational.**
*Recall that an irrational number is a real number which is not rational. You may use the fact that the rational numbers are closed under addition - if \(a\) and \(b\) are rational numbers, then \(a + b\) is rational as well.*
6. **Prove that if \(a\), \(b\), and \(c\) are positive real numbers with \(ab = c\), then \(a \leq \sqrt{c}\) or \(b \leq \sqrt{c}\).**
7. **Prove that**
\[
\sum_{i=1}^{n} i^3 = \frac{1}{4}n^2(n+1)^2
\]
for all positive integers \(n\).
8. **Prove that**
\[
\sum_{i=1}^{n} i \cdot i! = (n+1)! - 1
\]
for all positive integers \(n\).
9. **Prove that**
\[
2^n < n!
\]
for all positive integers \(n\) such that \(n \geq 4\).
Expert Solution
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 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
