Define a digit to be an element of the set (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) 00 00 Show that Σ n=1 10 = Σ n=1 - if and only if one of the following two conditions holds: 10 Either (i) a = b for every n or 12 (ii) there is some N&N U (0) such that • a = b_ for 1 ≤ n ≤N, a = b + 1, and a = 0 and b = 9 for all n > N.
Define a digit to be an element of the set (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) 00 00 Show that Σ n=1 10 = Σ n=1 - if and only if one of the following two conditions holds: 10 Either (i) a = b for every n or 12 (ii) there is some N&N U (0) such that • a = b_ for 1 ≤ n ≤N, a = b + 1, and a = 0 and b = 9 for all n > N.
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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![### Infinite Series and Equal Sums
**Definition**: A digit is defined as an element of the set \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).
**Objective**: Demonstrate that:
\[
\sum_{n=1}^{\infty} \frac{a_n}{10^n} = \sum_{n=1}^{\infty} \frac{b_n}{10^n}
\]
if and only if one of the following two conditions holds:
1. **Condition i**: \(a_n = b_n\) for every \(n\).
2. **Condition ii**: There exists some \(N\) in the set of natural numbers \(\mathbb{N} \cup \{0\}\) such that:
- \(a_n = b_n\) for \(1 \leq n \leq N\),
- \(a_{N+1} = b_{N+1} + 1\), and
- \(a_n = 0\) and \(b_n = 9\) for all \(n > N\).
This statement presents the conditions under which two infinite decimal expansions are equal. Condition ii accounts for cases involving terminal and repeating decimals that are mathematically equivalent (e.g., 0.999... = 1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2Fe92ee307-716c-49c9-b8a1-b3154d85397a%2Fhatvx6c_processed.png&w=3840&q=75)
Transcribed Image Text:### Infinite Series and Equal Sums
**Definition**: A digit is defined as an element of the set \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).
**Objective**: Demonstrate that:
\[
\sum_{n=1}^{\infty} \frac{a_n}{10^n} = \sum_{n=1}^{\infty} \frac{b_n}{10^n}
\]
if and only if one of the following two conditions holds:
1. **Condition i**: \(a_n = b_n\) for every \(n\).
2. **Condition ii**: There exists some \(N\) in the set of natural numbers \(\mathbb{N} \cup \{0\}\) such that:
- \(a_n = b_n\) for \(1 \leq n \leq N\),
- \(a_{N+1} = b_{N+1} + 1\), and
- \(a_n = 0\) and \(b_n = 9\) for all \(n > N\).
This statement presents the conditions under which two infinite decimal expansions are equal. Condition ii accounts for cases involving terminal and repeating decimals that are mathematically equivalent (e.g., 0.999... = 1).
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