Let (fn)nen be a sequence in C(0, 1]) and (ax)keN a sequence of positive numbers, such that the series 1 4% a converges and such that ISk+1 - Sell Sax for all k eN. Show that ISntk(2) – fn(z)| < ||fn+k – fall

Let (fn)nen be a sequence in C(0, 1]) and (ax)keN a sequence of positive numbers, such that the series 1 4% a converges and such that ISk+1 - Sell Sax for all k eN. Show that ISntk(2) – fn(z)| < ||fn+k – fall

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Description: Let (fn)nen be a sequence in C(0, 1]) and (ax)keN a sequence of positive numbers, suchthat the series 1 4% a converges and such thatISk+1 - Sell Sax for all k eN.Show thatISntk(2) – fn(z)| < ||fn+k – fall

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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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Let (fn)nen be a sequence in C(0, 1]) and (ax)keN a sequence of positive numbers, such
that the series 1 4% a converges and such that
ISk+1 - Sell Sax for all k eN.
Show that
ISntk(2) – fn(z)| < ||fn+k – fall <Eas
jun
for all n, k eN and z € (0, 1].
Use this to show that C([0, 1]) is complete.

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