Math 132 Test 4 4/16/08 Name_____________________________
Calculator–free
1-9. Determine if the following converge or diverge. Justify your response. For the series, mention the test that you
use to determine convergence or divergence. If it is a convergent sequence, give the limit to which it
converges. (6 pts. each)
1.
2
3
3 1
2 1
k
k
a
k k
2.
2
3
4
n
n
a
n
3.
0
3
!
k
kk
4.
1
7
( 1) 1
k
kk
5.
1
cos( )
k
k
7.
2
2
3 1
14 1
n
n
n
a
n n
8.
3k
k=1
k e
9.
2
3
3 1
2 1
k=1
k
k k
10. Determine which series converge. Find the sum of the series where possible. (12)
a)
55
2
5
4
5
8
5
16
...
b)
0 2 0 6 18 5 4 16 2. . . . . ...
c)
1
4
2
n
n
11. Find the interval of convergence for the power series:
1
( 1)
4
k
k
k
kx
.(10)
12 a. Find the first three non-zero terms of the Taylor series for f(x) = sin x about
x
. Show your work.
b. Use the first non-zero term of the polynomial to estimate sin 3. (12)
13. a. Find the first four non-zero terms of the Taylor series for f(x) = e3x about x = 0. Show your work.
b. Use your work above to find a polynomial approximation for g(x) = x2e3x.(12)
_______________________________________________________________________________________
1. C to 0; 2. C to 3; 3. C by ratio test; 4. C by alternating series test; 5. D kth term test; 7. D lim n->inf DNE;
8. C ratio test; 9. D limit comparison test compare to sum(1/k,k,0,inf), divergent p-series. 10 all Geometric
(a) a = 5, r = -0.5 converges to 10/3 (b) r = 3, diverges (c) r = ¼, a = 1/16 converges to 1/12
11. (-3, 5) 12a. -1(x-pi)+
3 5
1( ) ( ) / 3! ( ) / 5!x x x
; b) 0.14
13a)
3
2 3
0
1 3 (3 ) / 2! (3 ) / 3! (3 ) / !
k
k
x x x x k
b)
2 3 2 3 4 5
3 4.5 4.5
x
x e x x x x
