Find the erros : 0 = 0+o + 0+ %3D (1-1)+ (l-1) + (1-1)+ %3D ... 1-1+1-1 +1-1 +.. = I+(-1+1) +(-1+) + so ニ ..
Find the erros : 0 = 0+o + 0+ %3D (1-1)+ (l-1) + (1-1)+ %3D ... 1-1+1-1 +1-1 +.. = I+(-1+1) +(-1+) + so ニ ..
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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Find the error:
![### Mathematical Fallacy Exploration
**Objective:** Find the errors in the following mathematical steps.
1. Begin with an initial expression for zero:
\[
0 = 0 + 0 + 0 + \cdots
\]
2. Represent each zero as the sum of one and negative one:
\[
= (1 - 1) + (1 - 1) + (1 - 1) + \cdots
\]
3. Rearrange the terms:
\[
= 1 - 1 + 1 - 1 + 1 - 1 + \cdots
\]
4. Group the terms:
\[
= 1 + (-1 + 1) + (-1 + 1) + \cdots
\]
5. Simplify within the parentheses:
\[
= 1 + 0 + 0 + \cdots
\]
6. Conclude with an incorrect simplification:
\[
0 = 1
\]
---
**Explanation:** This sequence incorrectly combines and rearranges infinite series, leading to the fallacious conclusion that zero equals one. The error arises from inappropriate reassociation in an infinite series, disregarding the rules of convergence and permissible operations on divergent series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab56d24d-942e-4888-9f9d-899700a2a792%2Ff13c6155-9e5c-4a08-a33e-7eaa9e3a9461%2F42wcxue_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematical Fallacy Exploration
**Objective:** Find the errors in the following mathematical steps.
1. Begin with an initial expression for zero:
\[
0 = 0 + 0 + 0 + \cdots
\]
2. Represent each zero as the sum of one and negative one:
\[
= (1 - 1) + (1 - 1) + (1 - 1) + \cdots
\]
3. Rearrange the terms:
\[
= 1 - 1 + 1 - 1 + 1 - 1 + \cdots
\]
4. Group the terms:
\[
= 1 + (-1 + 1) + (-1 + 1) + \cdots
\]
5. Simplify within the parentheses:
\[
= 1 + 0 + 0 + \cdots
\]
6. Conclude with an incorrect simplification:
\[
0 = 1
\]
---
**Explanation:** This sequence incorrectly combines and rearranges infinite series, leading to the fallacious conclusion that zero equals one. The error arises from inappropriate reassociation in an infinite series, disregarding the rules of convergence and permissible operations on divergent series.
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