Download 4 Questions with Solutions of Calculus II - Exam 5 | MATH 142 and more Exams Calculus in PDF only on Docsity!
MATH 142- EXAM 5-April 22, 2005 Instructions. Justify answers for full credit. Calculators allowed. Time given: 60 minutes.
1.[12] Determine convergence/divergence for the following series. Jus- tify. โโ
n=
sin n 1 + n^2 | sin n| 1 + n^2 โค^
1 + n^2 โค^
n^2 โ^ CON V^ (comparison) โ^ โ
n=
(โ1)n^
n! 3 n |an+1| |an| =^
n + 1 3 โ โ โ^ DIV^ (ratio) โ^ โ
n=
(n + 1)^2 n^3 (n + 2)
(n + 1)^2 n^3 (n + 2) โผ^
n^2 n^4 =^
n^2 โ^ CON V^ (limit^ โ^ comparison) 2.[8] Find the number of terms you need to add to approximate each of the infinite sums below with |error| < 0 .01:
(a)
โ^ โ
n=
n^7 /^2
; (b)
โ^ โ
n=
(โ1)n^
n 4 n โซ (^) โ
N
dx x^7 /^2
=^25 N โ^5 /^2 < 10 โ^2 โ 2. 5 N 5 /^2 > 100 , N โฅ 5
N + 1
4 N^ +^
< 10 โ^2 , N = 4 works
3.[4] Find a representation of the function given below (choose one! ) as a power series at 0, including the radius of convergence:
(a)f (x) =
(x โ 2)^2 (b)f^ (x) =^
x x^2 + 4
[
2 โ x ]
โฒ =^1
2 [^
1 โ x 2 ]
โฒ =^1
โ^ โ
n=
xn 2 n^ )
โฒ =^1
โ^ โ
n=
nxnโ^1 2 n^ ,^ R^ = 2.
x x^2 + 4
= x 4
1 + x 42
= x 4
โ^ โ
n=
(โ x
2 4
)n^ =
โ^ โ
n=
(โ1)n^ x
2 n+ 4 n+^
R = 2.
4.[14] Use a power series to approximate to 5 decimal places the definite integral: (^) โซ
- 2 0
1 + x^5 dx
(4 steps: (i)[4] expand the integrand as a power series; (ii)[4] compute the definite integral, yielding an alternating series; (iii)[4]use the remainder esti- mate to compute the number N of terms needed; (iv) [2]compute the partial sum sN of the series to obtain the approximation.)
โซ (^0). 2
0
dx 1 + x^5 =
0
โ^ โ
n=
(โ1)nx^5 ndx =
โ^ โ
n=
(โ1)n^ x
5 n+ 5 n + 1 |
- 2 0 =
โ^ โ
n=
(โ1)n^ (0.2)
5 n+ 5 n + 1.
|rN | < (0.2)
5(N +1)+ 5(N + 1) + 1
5 N + 5 N + 6
< 10 โ^5 (N=1 works)
Approximate value: 0. 2 โ 0. 62 6 = 0. 1999893 ...
