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 $.

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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 ||

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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Question

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 $.

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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