Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

Could we prove the second question? (about the product)

Suppose \( f \) and \( g \) are \( C^n \) functions with Taylor expansions denoted \( T_n(f, x_0, x) \) and \( T_n(g, x_0, x) \). Prove that \( T_n(f, x_0, x) + T_n(g, x_0, x) \) is the Taylor expansion of \( f + g \) at \( x_0 \). Under the same hypotheses as exercise 11, show that the Taylor expansion of \( f \cdot g \) at \( x_0 \) is obtained by taking \( T_n(f, x_0, x) T_n(g, x_0, x) \) and retaining only the powers of \( (x - x_0) \) up to \( n \).
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Transcribed Image Text:Suppose \( f \) and \( g \) are \( C^n \) functions with Taylor expansions denoted \( T_n(f, x_0, x) \) and \( T_n(g, x_0, x) \). Prove that \( T_n(f, x_0, x) + T_n(g, x_0, x) \) is the Taylor expansion of \( f + g \) at \( x_0 \). Under the same hypotheses as exercise 11, show that the Taylor expansion of \( f \cdot g \) at \( x_0 \) is obtained by taking \( T_n(f, x_0, x) T_n(g, x_0, x) \) and retaining only the powers of \( (x - x_0) \) up to \( n \).
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