1 1 1 17. II П for every integer %3D 2i +1 2i+2 (2n +2)!’ n20.
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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Prove by induction
![**Problem 17:**
\[
\prod_{i=0}^{n} \left( \frac{1}{2i+1} \cdot \frac{1}{2i+2} \right) = \frac{1}{(2n+2)!}
\]
For every integer \( n \geq 0 \).
**Explanation:**
The problem presents a product notation \(\prod\) which is similar to summation but involves multiplying a sequence of terms. The expression inside the product involves two fractions, \(\frac{1}{2i+1}\) and \(\frac{1}{2i+2}\), for each integer \(i\) from 0 to \(n\).
The right side of the equation is the reciprocal of the factorial of \(2n+2\), denoted as \((2n+2)!\), which is the product of all positive integers up to \(2n+2\).
This equation asserts that the product of the sequence of terms on the left side equals the reciprocal of the factorial on the right side for any non-negative integer \(n\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2863cec8-c019-46a6-ad68-be87d1cedb5d%2F48584b3e-1be4-4cab-9606-31b70fe6d4a2%2Fzncllk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 17:**
\[
\prod_{i=0}^{n} \left( \frac{1}{2i+1} \cdot \frac{1}{2i+2} \right) = \frac{1}{(2n+2)!}
\]
For every integer \( n \geq 0 \).
**Explanation:**
The problem presents a product notation \(\prod\) which is similar to summation but involves multiplying a sequence of terms. The expression inside the product involves two fractions, \(\frac{1}{2i+1}\) and \(\frac{1}{2i+2}\), for each integer \(i\) from 0 to \(n\).
The right side of the equation is the reciprocal of the factorial of \(2n+2\), denoted as \((2n+2)!\), which is the product of all positive integers up to \(2n+2\).
This equation asserts that the product of the sequence of terms on the left side equals the reciprocal of the factorial on the right side for any non-negative integer \(n\).
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