Consider the sequence of functions {fn}n=2 where fr: [a, b] → R defined for n > 2 byn2xif 0

Definition: Suppose fn is a sequence of functions fn:S→ℝ and f:S→ℝ. Then fn→f pointwise on S if…

Answered: Consider the sequence of functions…

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Consider the sequence of functions {fn}=2 where fn: [a, b] → R defined for n 2 2 by if 0

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 72E

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Question

Consider the sequence of functions {fn}n=2 where fr: [a, b] → R defined for n > 2 by
n2x
if 0 <x<
fn
-n2 (x
if sxs1
a. Prove {fn}=2 converges pointwise to f(x) = 0.

Expert Solution

Step 1: Given.

Definition: Suppose $\{f_{n}\}$ is a sequence of functions $f_{n} : S \to ℝ$ and $f : S \to ℝ$ . Then $f_{n} \to f$ pointwise on $S$ if $f_{n} (x) \to f (x)$ as $n \to \infty$ for every $x \in A$ .

Given: $f_{n} : [a, b] \to ℝ$ defined for $n \geq 2$ by $f_{n} (x) = \{\begin{cases} n^{2} x, & if 0 \leq x \leq \frac{1}{n} \\ - n^{2} (x - \frac{2}{n}), & if \frac{1}{n} \leq x \leq \frac{2}{n} \\ 0, & if \frac{2}{n} \leq x \leq 1 \end{cases}$