Thomas Calculus 13th [Solutions]

728 Chapter 10 Infinite Sequences and Series
60. (a) s10
10
1
3
1.082036583; 1
b
b
4
dx lim x dx lim x lim 1 1 1
4
4 3
n
11 x
11 3 3 3993 3993
n 1
b b 11 b b
and
3
1
b
b
4
dx lim x dx lim x lim 1 1 1
10
4 3
x b 10 b
3
10 b 3b
3000 3000
1.082036583 1 s 1.082036583 1
3993 3000
1.08229 s 1.08237
(b) s 1 1.08229 1.08237 1.08233; error 1.08237 1.08229
4
n
2
2
n 1
0.00004
61. The total area will be
n
1 1 1 1 1
1
. The p-series converges to
n n n 1 2 n( n 1)
2
1 n 1 n
n 1 n
2
6 and
1
n
n( n
1
1)
converges to 1 (see Example 5). Thus we can write the area as the difference of these two values,
or
2
6
1 0.64493.
62. The area of the nth
trapezoid is
1 1 1 1 1 1 1 1
2 n n 1 n n 1 2 n ( n 1)
2 2
.
The total area will be
1 1 1 1 ,
2 2
2 2
n 1 n ( n 1)
since the series telescopes and has a value of 1.
10.4 COMPARISON TESTS
1
n
n 1
1. Compare with ,
2
which is a convergent p-series since p 2 1. Both series have nonnegative terms for
2 2 1 1
n n 30
n 1. For n 1, we have 30 .
2 2
n n Then by Comparison Test,
2
n 1 n
1
30
converges.
1
n
n 1
2. Compare with ,
3
which is a convergent p-series since p 3 1. Both series have nonnegative terms for
4 4 1 1 n n 1 n n 1
n n 2 n n 2 n n 2 n 2
n 1. For n 1, we have n n 2 . Then by Comparison
4 4 4 4 3 4 4
n
n
n 1
Test,
4
1
2
converges.
3. Compare with
1
,
n 2
n
which is a divergent p-series since p
1
1. Both series have nonnegative terms for
2
n 2. For n 2, we have n 1 n 1 1
. Then by Comparison Test, 1
n
1
n
n
2 n
1
diverges.
Copyright
2014 Pearson Education, Inc.