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12. Under the same hypotheses as exercise 11, show that the Taylor expansion of f.g at zo is obtained by taking Tn(f, xo, x)T,(9, xo, r) and retaining only the powers of (x – xo) up ton.

12. Under the same hypotheses as exercise 11, show that the Taylor expansion of f.g at zo is obtained by taking Tn(f, xo, x)T,(9, xo, r) and retaining only the powers of (x – xo) up ton.

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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Could you help me with just Exercise 12 (it uses Exercise 11's premises so I have attached that question as well)

**Exercise 11:** 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 \( x_0 \). **Exercise 12:** 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:**Exercise 11:** 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 \( x_0 \). **Exercise 12:** 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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