Compute the pro II 1 - k = 5 1 k
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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Please compute the product
![Certainly! Here is a transcription and explanation for the image:
---
**Mathematical Computation Exercise**
**Objective:** Compute the given product expression.
**Expression:**
\[
\prod_{k=5}^{5} \left( 1 - \frac{1}{k} \right) = 1 - \frac{1}{5}
\]
**Explanation:**
- The symbol \(\prod\) denotes a product similar to a summation, but for multiplication.
- The limits of the product are from \(k = 5\) to \(k = 5\). This implies that there is only one term in the sequence.
- The expression inside the product is:
\[
1 - \frac{1}{k}
\]
When \(k = 5\), this simplifies to:
\[
1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}
\]
**Solution:**
The result of the computation is:
\[
\frac{4}{5}
\]
This exercise demonstrates the evaluation of a simple product involving a single term. In cases where \(k\) ranges over more values, the process would involve multiplying all terms for \(k\) within the specified range.
---
This explanation is designed to aid in the understanding of product notation and basic evaluation for students studying mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2Fe795ef8d-a839-4f16-993c-4164ef2476b7%2F5sxjo0b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here is a transcription and explanation for the image:
---
**Mathematical Computation Exercise**
**Objective:** Compute the given product expression.
**Expression:**
\[
\prod_{k=5}^{5} \left( 1 - \frac{1}{k} \right) = 1 - \frac{1}{5}
\]
**Explanation:**
- The symbol \(\prod\) denotes a product similar to a summation, but for multiplication.
- The limits of the product are from \(k = 5\) to \(k = 5\). This implies that there is only one term in the sequence.
- The expression inside the product is:
\[
1 - \frac{1}{k}
\]
When \(k = 5\), this simplifies to:
\[
1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}
\]
**Solution:**
The result of the computation is:
\[
\frac{4}{5}
\]
This exercise demonstrates the evaluation of a simple product involving a single term. In cases where \(k\) ranges over more values, the process would involve multiplying all terms for \(k\) within the specified range.
---
This explanation is designed to aid in the understanding of product notation and basic evaluation for students studying mathematics.
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