Question 2* (Cauchy criterion) Recall that a sequence s, is a Cauchy sequence if for all > 0, there exists NER so that n, m> N ensures 8n - 8m < €. (a) Prove that the following is an equivalent definition of a Cauchy sequence: Sn is a Cauchy sequence if, for all e > 0, there exists NER so that n>m > N ensures Sn-8m| < €. (b) Prove the following theorem about series, known as the Cauchy criterion. THEOREM 2 (Cauchy Criterion). A series 1a is convergent if and only if for all e> 0 there exists NER so that n>m > N ensures n Σ k=m+1 (c) Now use the theorem you proved in part (a) to prove the following corollary: COROLLARY 3. If a series Ek-1 ak is convergent, then lim+∞o ak = 0. ake.
Question 2* (Cauchy criterion) Recall that a sequence s, is a Cauchy sequence if for all > 0, there exists NER so that n, m> N ensures 8n - 8m < €. (a) Prove that the following is an equivalent definition of a Cauchy sequence: Sn is a Cauchy sequence if, for all e > 0, there exists NER so that n>m > N ensures Sn-8m| < €. (b) Prove the following theorem about series, known as the Cauchy criterion. THEOREM 2 (Cauchy Criterion). A series 1a is convergent if and only if for all e> 0 there exists NER so that n>m > N ensures n Σ k=m+1 (c) Now use the theorem you proved in part (a) to prove the following corollary: COROLLARY 3. If a series Ek-1 ak is convergent, then lim+∞o ak = 0. ake.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Question 2* (Cauchy criterion)
Recall that a sequence sn is a Cauchy sequence if
for all > 0, there exists NER so that n, m> N ensures 8n - 8m| < €.
(a) Prove that the following is an equivalent definition of a Cauchy sequence:
Sn is a Cauchy sequence if, for all e > 0, there exists NER so that n>m > N ensures Sn-Sm < €.
(b) Prove the following theorem about series, known as the Cauchy criterion.
THEOREM 2 (Cauchy Criterion). A series 1 ak is convergent if and only if
for all e> 0 there exists NER so that n>m > N ensures Σ
k=m+1
(c) Now use the theorem you proved in part (a) to prove the following corollary:
COROLLARY 3. If a series 1 ak is convergent, then lim+∞ak = 0.
(Hint: take n = m + 1 in the theorem from part (a).)
ak E.
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