ΣΣΣ k i=4 (j=i k=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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Question
Calculate the summation
![The image contains the following mathematical expression involving nested summations:
\[
\sum_{i=4}^{n} \left( \sum_{j=i}^{n} \left( \sum_{k=1}^{j} k \right) \right)
\]
### Explanation:
This is a triple summation. Here's a breakdown of each component:
1. **Outer Sum**:
- \(\sum_{i=4}^{n}\): This sums over \(i\) starting from 4 up to \(n\).
2. **Middle Sum**:
- \(\sum_{j=i}^{n}\): For each \(i\), this sums over \(j\) starting from \(i\) up to \(n\).
3. **Innermost Sum**:
- \(\sum_{k=1}^{j} k\): For each \(j\), this sums over \(k\) starting from 1 up to \(j\).
The entire expression calculates the cumulative sum of these nested sequences. Such operations are commonly used in combinatorics and computational algorithms to evaluate series and perform multi-dimensional summing.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc5ac1678-3f3b-440a-b3b5-846319a7a56a%2Ff36f2ce8-c37f-47e9-b0fa-3b95953a316e%2Fk7bbva_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains the following mathematical expression involving nested summations:
\[
\sum_{i=4}^{n} \left( \sum_{j=i}^{n} \left( \sum_{k=1}^{j} k \right) \right)
\]
### Explanation:
This is a triple summation. Here's a breakdown of each component:
1. **Outer Sum**:
- \(\sum_{i=4}^{n}\): This sums over \(i\) starting from 4 up to \(n\).
2. **Middle Sum**:
- \(\sum_{j=i}^{n}\): For each \(i\), this sums over \(j\) starting from \(i\) up to \(n\).
3. **Innermost Sum**:
- \(\sum_{k=1}^{j} k\): For each \(j\), this sums over \(k\) starting from 1 up to \(j\).
The entire expression calculates the cumulative sum of these nested sequences. Such operations are commonly used in combinatorics and computational algorithms to evaluate series and perform multi-dimensional summing.
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