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Prove that the Taylor series around a f(x) for all x e R. * for f(x) cos(x) converges to ||

Prove that the Taylor series around a f(x) for all x e R. * for f(x) cos(x) converges to ||

Advanced Engineering Mathematics
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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COURSE: Mathematical/Real Analysis (TS4)

TOPIC: Taylor Series

 

### Problem Statement **Objective**: Prove that the Taylor series around \( a = \frac{\pi}{2} \) for \( f(x) = \cos(x) \) converges to \( f(x) \) for all \( x \in \mathbb{R} \). ### Details - **Function**: \( f(x) = \cos(x) \) - **Point of Expansion**: \( a = \frac{\pi}{2} \) - **Domain**: All real numbers \( x \in \mathbb{R} \) The task is to show that when we express \( \cos(x) \) using its Taylor series at \( a = \frac{\pi}{2} \), the series converges to the actual value of \(\cos(x)\) for any real number \( x \).
Transcribed Image Text:### Problem Statement **Objective**: Prove that the Taylor series around \( a = \frac{\pi}{2} \) for \( f(x) = \cos(x) \) converges to \( f(x) \) for all \( x \in \mathbb{R} \). ### Details - **Function**: \( f(x) = \cos(x) \) - **Point of Expansion**: \( a = \frac{\pi}{2} \) - **Domain**: All real numbers \( x \in \mathbb{R} \) The task is to show that when we express \( \cos(x) \) using its Taylor series at \( a = \frac{\pi}{2} \), the series converges to the actual value of \(\cos(x)\) for any real number \( x \).
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