1. Fix R € (0,1). Prove that oth converges pointwise to f(t) = ₁¹t on S = [-R, R]. Hint: Fix R € (0, 1) and t = [-R, R]. Let sn(t) to +²+...+1. Prove that limno Sn(t)= by rewriting sn(t) as a certain fraction. 1-t -
1. Fix R € (0,1). Prove that oth converges pointwise to f(t) = ₁¹t on S = [-R, R]. Hint: Fix R € (0, 1) and t = [-R, R]. Let sn(t) to +²+...+1. Prove that limno Sn(t)= by rewriting sn(t) as a certain fraction. 1-t -
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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![**Problem 1:**
Fix \( R \in (0, 1) \). Prove that \(\sum_{n=0}^\infty t^n\) converges pointwise to \( f(t) = \frac{1}{1-t} \) on \( S = [-R, R] \).
**Hint:** Fix \( R \in (0, 1) \) and \( t \in [-R, R] \). Let \( s_n(t) = t^0 + t^1 + \cdots + t^{n-1} \). Prove that \(\lim_{n \to \infty} s_n(t) = \frac{1}{1-t}\) by rewriting \( s_n(t) \) as a certain fraction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2Ff9b69110-3178-45d0-9256-8a05e08e7fc9%2Fcrdff8p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1:**
Fix \( R \in (0, 1) \). Prove that \(\sum_{n=0}^\infty t^n\) converges pointwise to \( f(t) = \frac{1}{1-t} \) on \( S = [-R, R] \).
**Hint:** Fix \( R \in (0, 1) \) and \( t \in [-R, R] \). Let \( s_n(t) = t^0 + t^1 + \cdots + t^{n-1} \). Prove that \(\lim_{n \to \infty} s_n(t) = \frac{1}{1-t}\) by rewriting \( s_n(t) \) as a certain fraction.
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