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 \( n \).

We will use the formula for the product of power series : And the Lebinitz rure of the n-th…

Answered: 12. Under the same hypotheses as…

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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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.