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HiPlease check the attached file for details, let me know if you have any questions,in case that you need, I also attached pdf file, thank you. James
1.
1
A
Bx C
2
( x 1)( x 3) ( x 3) ( x 1)
2
1
( Bx C )
A 2
( x 3)
( x 1)
( x 1)
2
Let x =3,
A = 1/10
1
1 ( Bx C )
2
( x 3)
( x 1) 10 ( x 1)
2
x2 1
Bx 2 (3B C ) x 3C
10
1
B
10
3
C
10
1
1
1
1
x3
dx (
2
)dx
( x 1)( x 3)
10
( x 3) ( x 1)
2
1
1
3
ln | x 3 | ln( x 2 1) tan 1 x C
10
20
10
2. a) limit of an
Diverges. Because
|an| = 3n/n2 diverges to infinity at n = infinity.
b) Ratio test
Converges . Since
an1
e n1 (n 1) 3
lim
n a
n
e n n 3
n
lim
1
lim e 1 (1 ) 3
n
n
1/ e 1
c)Ratio test.
Converges. Since
an1
2 ( n1) ( n 1) 2
lim
n a
n
2 n n 2
n
lim
1
lim 2 1 (1 ) 2
n
n
1/ 2 1
d) Integration test.
Diverges. Since
1
x ln x dx ln(ln
x)
1
1
does not converge.
e) Comparison test.
Converges. Since
|
sin n 1 1
1
2
| 2 2 2
2
n
n
n
n
and
2
n
n 1
2
Converges.
3.
1
x(ln x)
p
dx
1
d ln x
(ln x) p
ln ln x, p 1
1
1
p 1
,
p
1
1 p (ln x)
For the integral converge, it needs to require that p-1>0, or p>1. Then the integral converge to
1
(...