………………………………………………………………………………………
I. Fill the blanks。
1. If the series 1
1 ( 1)n
p n n
=
− absolutely converges, then p satisfies ;
2. The convergence set of 0
1 ( 3)
3
n
n n
x
=
− is
3. The graph of + +
= x
dt t
t xf
0 21
1 )( is concave up on the interval _____________
4. 2
1
21
sin 1
1
x x dx
x−
+ =
+ =___________.
II. Calculations
1. 0
(1 cos 2 ) lim
tan sinx
x x
x x→
−
−
2. x x
x
)(coslim 0+→
3. arctanx xdx
4.1 2
2
1
1x dx
x
−
4.2 2
99
0 sin x dx
4.3 20
cos
1 sin
x x dx
x
+
5. Find +
−= 0
dxexI xnn , where n is a natural number.
6. Suppose that R is bounded by 3, 2, 0y x x y= = = .Find the volumes of the revolution
solids that are obtained by revolving R about x axis and y axis respectively. And the
revolution’s side area with expression.
7. Test the convergence of series ( )
( ) 3
1
cos 2 1 1
2nn
n n a
a
=
+
+
8. Find , 其中 .
III Applications.
1. The diagram represents an equilateral triangle containing infinitely many circles, tangent to the triangle and to neighboring circles, and reaching into the corners. What fraction of the area of the triangle is occupied by the circles?
2. Expand into power series and find .