2.1 Basic Concepts: Convergence and Limits 55The latter inequality will be true if9n - 21
23 > 2,000
9n - 21 > 46,000
9n > 46,021
n > 5, 113.444 · · ·.Take no = 5, 114. We have shown that n 2 5, 114 ==? --- - < .001.
I2n + 3 21
3n- 7 3
That is, when n 2 5, 114 , Xn will approximate lim Xn to three decimal places.
n->oo(c) Let c > 0 be a fixed but arbitrary positive number. We want to find an
I
no EN 3 n 2 no==? 2n +^3 21
3
n _
7- 3 < E:. As shown above, if n 2 3,
I
2n + 3 21 23
3n - 7 - 3 = 9n - 21°Thus, our objective is to find an no E N 3 no 2 3 and23
n2no=? <E:.
.--{.) 9n -^21
1pl"'-
t\G V1 Notice that when n > 3,
23 27
gn
21
< gn
21
(since 23 < 27 and 9n - 21 > 0)
<^27 ( since - 21 > -27 ==? 9n - 21 > 9n - 27
9n- 273==? 1 < __ 1_)
9n - 21 9n - 27
<
n-3
23
Thus, when n > 3, < E:
9n - 21if _
3
_ · < E:. But the last inequality will
n-3
be guaranteed ifn-3
31
> -3
i.e., n - 3 > -
€3
i.e., n > -+ 3.
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