Prove by induction that Vn € {0, 1, 2, ...} and for any a + 1, =(n + 1)an + nan+1 (1-a)² 72 Σκακ k=1 k−1 = +1
Prove by induction that Vn € {0, 1, 2, ...} and for any a + 1, =(n + 1)an + nan+1 (1-a)² 72 Σκακ k=1 k−1 = +1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you help me with these please? Thanks.
![**Mathematical Induction Problems**
1. **Proof by Induction for a Series Formula:**
Prove by induction that for every \( n \in \{0, 1, 2, \ldots\} \) and for any \( \alpha \neq 1 \),
\[
\sum_{k=1}^{n} k \alpha^{k-1} = \frac{-(n+1)\alpha^n + n\alpha^{n+1} + 1}{(1-\alpha)^2}.
\]
**Explanation:**
This problem involves proving a formula for the sum of the series where each term is of the form \( k \alpha^{k-1} \). The series runs from \( k = 1 \) to \( n \). The target expression is a function of \( n \) and \( \alpha \), involving terms up to \( \alpha^n \).
2. **Proof by Induction for an Integral Formula:**
Prove by induction that for all \( k \in \mathbb{N} \),
\[
\int_{0}^{\infty} x^{k-1} e^{-x} \, dx = (k-1)!.
\]
**Explanation:**
This problem requires proving a formula for the improper integral of the function \( x^{k-1} e^{-x} \). The expression on the right-hand side, \( (k-1)! \), is the factorial of \( k-1 \), indicating the relationship to the Gamma function, where the integral of this form equals the factorial of one less than the power of \( x \).
These problems blend induction with calculus and algebra, illustrating important mathematical techniques and properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72f9ec78-21fa-4141-958a-3b1eb6c54d04%2F6cc0dc0f-f74b-4de4-95dc-580f19b38a6e%2F6vgajra_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Induction Problems**
1. **Proof by Induction for a Series Formula:**
Prove by induction that for every \( n \in \{0, 1, 2, \ldots\} \) and for any \( \alpha \neq 1 \),
\[
\sum_{k=1}^{n} k \alpha^{k-1} = \frac{-(n+1)\alpha^n + n\alpha^{n+1} + 1}{(1-\alpha)^2}.
\]
**Explanation:**
This problem involves proving a formula for the sum of the series where each term is of the form \( k \alpha^{k-1} \). The series runs from \( k = 1 \) to \( n \). The target expression is a function of \( n \) and \( \alpha \), involving terms up to \( \alpha^n \).
2. **Proof by Induction for an Integral Formula:**
Prove by induction that for all \( k \in \mathbb{N} \),
\[
\int_{0}^{\infty} x^{k-1} e^{-x} \, dx = (k-1)!.
\]
**Explanation:**
This problem requires proving a formula for the improper integral of the function \( x^{k-1} e^{-x} \). The expression on the right-hand side, \( (k-1)! \), is the factorial of \( k-1 \), indicating the relationship to the Gamma function, where the integral of this form equals the factorial of one less than the power of \( x \).
These problems blend induction with calculus and algebra, illustrating important mathematical techniques and properties.
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