Let $ I $ be a bounded interval, and $ Z $ a Banach space. Let $ (f_n) $ be a sequence of differentiable functions $ f_n : I \to Z $ such that $ \sum_{n=1}^\infty f_n(x^\circ) $ converges to $ Z $ for some $ x^\circ \in I $ and there exists a sequence of positive numbers $ (M_n) $ such that $ \|f'_n(x)\| \leq M_n $ for all $ n \in \mathbb{N} $ and $ x \in I $ and $ \sum_{n=1}^\infty M_k $ converges in $ \mathbb{R} $.Show that $ \sum_{n=1}^\infty f_n(x) $ converges uniformly for $ x \in I $ and provides a differentiable function $ f : I \to Z $ such that $ f'(x) = \sum_{n=1}^\infty f'_n(x) $, with the convergence of the latter series being also uniform for $ x \in I $.

Name: Let $ I $ be a bounded interval, and $ Z $ a Banach space. Let $
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Description: Let \( I $ be a bounded interval, and $ Z $ a Banach space. Let $ (f_n) $ be a sequence of differentiable functions $ f_n : I \to Z $ such that $ \sum_{n=1}^\infty f_n(x^\circ) $ converges to $ Z $ for some $ x^\circ \in I $ and there e

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let $ I $ be a bounded interval, and $ Z $ a Banach space. Let $ (f_n) $ be a sequence of differentiable functions $ f_n : I \to Z $ such that $ \sum_{n=1}^\infty f_n(x^\circ) $ converges to $ Z $ for some $ x^\circ \in I $ and there exists a sequence of positive numbers $ (M_n) $ such that $ \|f'_n(x)\| \leq M_n $ for all $ n \in \mathbb{N} $ and $ x \in I $ and $ \sum_{n=1}^\infty M_k $ converges in $ \mathbb{R} $.

Show that $ \sum_{n=1}^\infty f_n(x) $ converges uniformly for $ x \in I $ and provides a differentiable function $ f : I \to Z $ such that $ f'(x) = \sum_{n=1}^\infty f'_n(x) $, with the convergence of the latter series being also uniform for $ x \in I $.

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Step 1

given

let $I$ be a bounded interval and $Z$ a banach space . let $f_{n}$ be a sequence of differentiable functions $f_{n} : I \to Z$ such that $\sum_{n = 1}^{\infty} f_{n} (x^{⋄})$ converges to $Z$ for some $x^{⋄} \in I$ and there exist a sequence of positive numbers $(M_{n})$ such that $||f_{n}^{'} (x)|| \leq M_{n} \forall x \in I and \sum_{n = 1}^{\infty} M_{k} converges in ℝ$

to show

$\sum_{n = 1}^{\infty} f_{n} (x) converges uniformly for x \in I$ and provides a differentiable function $f : I \to Z$ such that $f^{'} (x) = \sum_{n = 1}^{\infty} f_{n}^{'} (x)$

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