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Which of the following statements imply that {an} converges to a? (a) For every integer m > 0, there is an integer N > 0 such that an - al when n > N. (b) For each 0 < ɛ < 1, there is an integer N> 0 such that an - a| < 3ɛ when n > N². (c) For each 0 < ɛ < 1, there is an integer N> 0 such that an - a| < ɛ¯¹ when n > N. (d) For each N > 0, there is ɛ > 0 such that an - a| < N−¹ when n > N +ɛ. (e) For each > 0, there is an integer N> 0 such that an - a| < ɛ¯¹ when n > N². (f) For each > 0, there is an integer N> 0 such that an - a| < ɛ when n = N + k for all positive integer k. m (g) For each & > 0, there is an integer N> 0 such that an - a| <ɛ when n = N +2k for all positive integer k.

Which of the following statements imply that {an} converges to a? (a) For every integer m > 0, there is an integer N > 0 such that an - al when n > N. (b) For each 0 < ɛ < 1, there is an integer N> 0 such that an - a| < 3ɛ when n > N². (c) For each 0 < ɛ < 1, there is an integer N> 0 such that an - a| < ɛ¯¹ when n > N. (d) For each N > 0, there is ɛ > 0 such that an - a| < N−¹ when n > N +ɛ. (e) For each > 0, there is an integer N> 0 such that an - a| < ɛ¯¹ when n > N². (f) For each > 0, there is an integer N> 0 such that an - a| < ɛ when n = N + k for all positive integer k. m (g) For each & > 0, there is an integer N> 0 such that an - a| <ɛ when n = N +2k for all positive integer k.

Advanced Engineering Mathematics
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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(11) Which of the following statements imply that {an} converges to a? (a) For every integer m > 0, there is an integer N > 0 such that |an − a| < when m n > N. - ε (b) For each 0 < ɛ < 1, there is an integer N> 0 such that an - a| < 3ɛ when n > N². (c) For each 0 < ɛ < 1, there is an integer N >0 such that an a| < -¹ when n > N. (d) For each N > 0, there is ɛ > 0 such that an - a| < N-¹ when n > N + ε. N−¹ (e) For each > 0, there is an integer N> 0 such that an - a < ɛ¯¹ when n > N². (f) For each > 0, there is an integer N> 0 such that all positive integer k. -1 an - a < & when n = N + k for (g) For each & > 0, there is an integer N> 0 such that an − a| < ɛ when n = N +2k for all positive integer k.
Transcribed Image Text:(11) Which of the following statements imply that {an} converges to a? (a) For every integer m > 0, there is an integer N > 0 such that |an − a| < when m n > N. - ε (b) For each 0 < ɛ < 1, there is an integer N> 0 such that an - a| < 3ɛ when n > N². (c) For each 0 < ɛ < 1, there is an integer N >0 such that an a| < -¹ when n > N. (d) For each N > 0, there is ɛ > 0 such that an - a| < N-¹ when n > N + ε. N−¹ (e) For each > 0, there is an integer N> 0 such that an - a < ɛ¯¹ when n > N². (f) For each > 0, there is an integer N> 0 such that all positive integer k. -1 an - a < & when n = N + k for (g) For each & > 0, there is an integer N> 0 such that an − a| < ɛ when n = N +2k for all positive integer k.
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