The following sequence, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.a. Examine the first three terms of the sequence to determine whetherthe sequence is nondecreasing or nonincreasing.b. Use analytical methods to find the limit of the sequence.an + 1 = (1)/(2) (an + (2)/(an)); a0 = 2 THEOREM 10.5 Bounded Monotonic SequenceA bounded monotonic sequence converges.Figure 10.21 shows the two cases of this theorem. In the first case, we see a nondecreasingsequence, all of whose terms are less than M. It must converge to a limit less than or equalto M. Similarly, a nonincreasing sequence, all of whose terms are greater than N, mustconverge to a limit greater than or equal to N.MNonincreasingbounded belowNondecreasingbounded above1520101520 n(a)(b)Figure 10.21

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Description: The following sequence, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.a. Examine the first three terms of the sequence to determine whetherthe sequence is nondecreasing or nonincreasing.b. Use ana

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Answered: THEOREM 10.5 Bounded Monotonic Sequence…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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The following sequence, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.
a. Examine the first three terms of the sequence to determine whether
the sequence is nondecreasing or nonincreasing.
b. Use analytical methods to find the limit of the sequence.

a_n_{+ 1} = (1)/(2) (a_n + (2)/(a_n)); a₀ = 2

THEOREM 10.5 Bounded Monotonic Sequence
A bounded monotonic sequence converges.
Figure 10.21 shows the two cases of this theorem. In the first case, we see a nondecreasing
sequence, all of whose terms are less than M. It must converge to a limit less than or equal
to M. Similarly, a nonincreasing sequence, all of whose terms are greater than N, must
converge to a limit greater than or equal to N.
M
Nonincreasing
bounded below
Nondecreasing
bounded above
15
20
10
15
20 n
(a)
(b)
Figure 10.21

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