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Suppose f and g are C" functions with Taylor expansions denoted Ta(f, xo, x) and T,(9, xo, x). Prove that T(f,x0, x)+ Tn(9, co, x) is the Taylor expansion of f + g at xo.

Suppose f and g are C" functions with Taylor expansions denoted Ta(f, xo, x) and T,(9, xo, x). Prove that T(f,x0, x)+ Tn(9, co, x) is the Taylor expansion of f + g at xo.

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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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 \( n \).
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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