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Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Explain: e is really the limit of a sequence, and the value usually stated, 2.71828… is a numerical approximation of that limit.
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- One of the most important definitions in mathematics (calculus in particular) is that of the limit of a sequence. Consider the infinite sequence:1,1/2,1/3,1/4,.... Such a sequence can be represented as follows: an=1/n , n ∈ N Note that the elements of the sequence get arbitrarily close to 0 as n→∞. We call such a point the limit of the sequence. The formal definition for “L is the limit of a sequence an” is as follows: ∀∈ > 0, ∃N ∈ N, ∀ n ∈ N, n > N=⇒ |an−L| < ∈Negate this proposition to obtain the definition of “L is NOT the limit of a sequence an.”arrow_forward1. Compare Arithmetic sequences to Geometric Sequences.arrow_forward1. State the limit of the graph below that represents a sequence. 60 50 a) 0 b) – 1.2 c) 8 d) +∞ 40 30 20 10 34 567 8 9%arrow_forward
- In the second sequence, between the second to last line, it looks as though an extra n was pulled out of the limit. Why is that?arrow_forwardI need help with thatarrow_forward%24 2. Define a sequence {a,}1 by -{(--2). An+1 ɑj = 10. an First, note that a, > 0 for all n. Answer the following questions. (a) Explain why a, 2 /2 for all n. (b) Explain why {an}1 is non-increasing (that is, monotonically decreasing). (c) Explain why the limit lim a, exists, and find the limit. 112arrow_forward
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