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Find two mistakes in the following 'proof' and give a counterexample that shows wh the claim can't be true. Consider a sequence (an) such that for all e > 0 we can find N EN such that |an| < E. Define the corresponding series oan. Let SN = no an give the sequence of partial sums. Because of the assumption on the sequence (an), we can see that there is an N EN such that |SN+1 - SN| = |an| < E. Since this is true for all ɛ, the sequence (SN) is Cauchy, and hence the sequence of partial sums converge, and so the series is summable.

Find two mistakes in the following 'proof' and give a counterexample that shows wh the claim can't be true. Consider a sequence (an) such that for all e > 0 we can find N EN such that |an| < E. Define the corresponding series oan. Let SN = no an give the sequence of partial sums. Because of the assumption on the sequence (an), we can see that there is an N EN such that |SN+1 - SN| = |an| < E. Since this is true for all ɛ, the sequence (SN) is Cauchy, and hence the sequence of partial sums converge, and so the series is summable.

Algebra & Trigonometry with Analytic Geometry
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 24E
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Find two mistakes in the following 'proof' and give a counterexample that shows why the claim can't be true. Σwo an give Consider a sequence (an) such that for all e > 0 we can find NEN such that an < E. Define the corresponding series oan. Let SN the sequence of partial sums. Because of the assumption on the sequence (an), we can see that there is an N EN such that |SN+1 - SN| = |an| < E. Since this is true for all e, the sequence (SN) is Cauchy, and hence the sequence of partial sums converge, and so the series is summable. =
Transcribed Image Text:Find two mistakes in the following 'proof' and give a counterexample that shows why the claim can't be true. Σwo an give Consider a sequence (an) such that for all e > 0 we can find NEN such that an < E. Define the corresponding series oan. Let SN the sequence of partial sums. Because of the assumption on the sequence (an), we can see that there is an N EN such that |SN+1 - SN| = |an| < E. Since this is true for all e, the sequence (SN) is Cauchy, and hence the sequence of partial sums converge, and so the series is summable. =
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