Thomas Calculus 13th [Solutions]

Section 10.1 Sequences 703
40.
1
n
n
( 1)
lim lim 0
2 n
n
n 2
converges
41. lim 2n
lim 2n
lim 2 2
n 1 n 1 1
1
n n n
n
converges
42. lim 1 lim 10
n
n (0.9) n
9
n
diverges
43. lim sin 1 1
2 n
sin lim 2 n
sin 2
1
n
n
converges
44. lim n cos ( n ) lim ( n )( 1) n does not exist diverges
n
n
45. lim sin n 0
n
n
because
1 sin n 1
n n n
converges by the Sandwich Theorem for sequences
46.
lim sin n 0
n
n 2
2
because
2
0 sin 1
n n
n converges by the Sandwich Theorem for sequences
2 2
47. lim n lim 1 0
n
n
n 2 n 2 ln 2
converges
(using l'Hopital's rule)
48.
n n 2 n 3
n 3 ln 3 3 (ln 3) 3 (ln 3)
lim 3 lim lim lim
3 2
n n n 3n
n
6n
n
6
diverges
(using l'Hopital's rule)
49.
1
n 1
1
2 n
2
n
1
n
ln ( n 1) 2 n
lim lim lim lim 0
n n n n
n 1
n 1
converges
50.
1
n
2
2n
ln n
lim lim 1
n
ln 2n
n
converges
51.
1 n
lim 8 1
n
converges (Theorem 5, #3)
52.
1 n
lim (0.03) 1
n
converges (Theorem 5, #3)
53.
lim 1
n
7
n
n
7
e converges (Theorem 5, #5)
54.
n
lim 1 1 lim 1
n
n
n
( 1)
n
1
e converges (Theorem 5, #5)
n
55.
lim
n
10n 1 n
lim 10
1 n
n 1 1 1 converges (Theorem 5, #3 and #2)
n
n
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