Calculus- Early Transcendentals, 2021a

9.1. Sequences 341
Example 9.8: Convergence of a Rational Fraction
{ n ∞
Determine whether
converges or diverges. If it converges, compute the limit.
n + 1}
n=0
Solution. Defining f (x)=
Thus the sequence converges to 1.
x+1 x we obtain
lim
x→∞
x
x + 1 = lim x→∞ 1 − 1
x + 1 = 1 − 0 = 1.
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Example 9.9: Convergence of Ratio with Natural Logarithm
{ lnn
Determine whether converges or diverges. If it converges, compute the limit.
n
} ∞
n=1
Solution. We compute
lnx
lim
x→∞ x
= lim 1/x
x→∞ 1 = 0,
using L’Hôpital’s Rule. Thus the sequence converges to 0.
♣
Example 9.10: Alternating Ones
Determine whether {(−1) n } ∞ n=0 converges or diverges. If it converges, compute the limit.
Solution. f (x) =(−1) x is undefined for irrational values of x so lim x→∞ (−1) x does not exist. However,
the sequence has a very simple pattern:
1,−1,1,−1,1...
and clearly diverges.
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Example 9.11: Convergence of Exponential
Determine whether {(−1/2) n } ∞ n=0 converges or diverges. If it converges, compute the limit.
Solution. We consider the sequence {|(−1/2) n |} ∞ n=0 = {(1/2)n } ∞ n=0 .Then
) 1 x
1
lim = lim
x→∞(
2 x→∞ 2 x = 0,
so by Theorem 9.7 the sequence converges to 0.
♣