Show that lim (1 – 2²)"dr 0, n 00 0,
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
1.
![**Problem:**
Show that
\[
\lim_{n \to \infty} \int_{0}^{1} (1-x^2)^n \, dx = 0
\]
without attempting to evaluate the integral explicitly.
---
**Explanation:**
The problem requires demonstrating that the limit of the integral approaches zero as \( n \) tends to infinity, focusing on a mathematical analysis rather than direct evaluation of the integral.
**Approach:**
1. **Understanding the Function**: The function \((1-x^2)^n\) represents a curve where the base, \(1-x^2\), is a parabola. As \( n \) increases, this function gets steeper and narrower around \( x = 0 \).
2. **Behavior as \( n \to \infty \)**: Intuitively, as \( n \) becomes very large, the function \((1-x^2)^n\) approaches 0 for all \( x \) in \((0, 1)\) except near \( x = 0\). This is because \((1-x^2)\) is less than one in the interval \((0, 1)\), and raising it to the power \( n \) causes it to approach zero.
3. **Limit of the Integral**: The integral represents the area under the curve \((1-x^2)^n\) from 0 to 1. As the function gets steeper and narrower at \( x = 0 \) while approaching 0 elsewhere in the interval, the area under the curve effectively shrinks to zero.
This mathematical exercise reveals behaviors of functions and their integrals, emphasizing limits and continuity without evaluation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87bd9bd0-40fd-4172-a50a-abb52eb6a8c1%2Fa70d6900-73ff-4df3-abfb-2852a8f29356%2Fxpzodmj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Show that
\[
\lim_{n \to \infty} \int_{0}^{1} (1-x^2)^n \, dx = 0
\]
without attempting to evaluate the integral explicitly.
---
**Explanation:**
The problem requires demonstrating that the limit of the integral approaches zero as \( n \) tends to infinity, focusing on a mathematical analysis rather than direct evaluation of the integral.
**Approach:**
1. **Understanding the Function**: The function \((1-x^2)^n\) represents a curve where the base, \(1-x^2\), is a parabola. As \( n \) increases, this function gets steeper and narrower around \( x = 0 \).
2. **Behavior as \( n \to \infty \)**: Intuitively, as \( n \) becomes very large, the function \((1-x^2)^n\) approaches 0 for all \( x \) in \((0, 1)\) except near \( x = 0\). This is because \((1-x^2)\) is less than one in the interval \((0, 1)\), and raising it to the power \( n \) causes it to approach zero.
3. **Limit of the Integral**: The integral represents the area under the curve \((1-x^2)^n\) from 0 to 1. As the function gets steeper and narrower at \( x = 0 \) while approaching 0 elsewhere in the interval, the area under the curve effectively shrinks to zero.
This mathematical exercise reveals behaviors of functions and their integrals, emphasizing limits and continuity without evaluation.
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