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Unformatted text preview: MATH 162 * SPRING 2010 — THIRD EXAM — APRIL 13, 2010
VERSION 01
MARK TEST NUMBER 01 ON YOUR SCANTRON STUDENT NAME STUDENT ID RECITATION INSTRUCTOR INSTRUCTOR RECITATION TIME INSTRUCTIONS 1. Fill in all the information requested above and the version number of the test on your
scantron sheet. 2. This booklet contains 13 problems. Problem 1 is worth 4 points. The others are worth
8 points each. The maximum score is 100 points. 3. For each problem mark your answer on the scantron sheet and also circle it is this
booklet. 4. Work only on the pages of this booklet.
5. Books, notes and calculators are not allowed.
6. At the end turn in your exam and scantron sheet to your recitation instructor. Useful Formulas m 0° $2n+l
: _ n ' = 1 n__._
cosx 1) any sma: ;( ) (2n+1)! 2 00
1)(4 points) For the series 2 (—1)”n2, the partial sum 84 equals 71,21 A) 2.
B) 10. C) 10. 2)(8 points) Which of the following statements are true? (I) If lim an 2 0, then 2 an converges. 77r~>00
n21 (II) If 2 lan converges, then 2 an converges. 1 71:1 (HI) If f: n=1 00
converges, then E an converges. n=1 an+1
an (IV) If 0 3 an S b" and Z bn diverges, then 2 a71 diverges.
n=1 n=l (V) If lirn 5%” = 2, then 2 an converges. n—yoo A) (I), (II) and (III) only. n=1 B)(I), (II) and (IV) only.
C)(II), (IV) and (V) only.
D)(II), (III) and (V) only. E)(II), (III) and (IV) only. °° 1
3)(8 points) Which of the following alternatives is true about the series Z W? 71:2 00
1
A) It converges by the comparison test with Z 71:1 °° 1
B) It diverges by the comparison test with Z 71:1 00
1
C) It converges by the comparison test with Z 71:1 °° 1
D) It diverges by the comparison test with Z 71:1 E) It converges by the integral test. 4)(8 points) Which of the following series diverge?
n + 1
(1) 2 n2
71:1
00 712 + 71
(11) 2
71:1 712—71 (111) Z 1—H}; 71:2 A) (I) only. B)(II) only.
C)(I) and (II) only.
D)(II) and (III) only. E)All of them. 4 5)(8 points) Which statement is true about the following series? A)All are conditionally convergent.
B)All are divergent.
C)(I) is conditionally convergent; (H) is absolutely convergent. D)(I) is absolutely convergent; (II) is conditionally convergent. and (II) are conditionally convergent; (III) is divergent. 6)(8 points) Let S : i: Find the smallest integer N such that we
can be sure that SN — S] < {(176, Where SN 2 i (ﬂag—7:233
n=1
A) 8
B) 9
C) 10
D) 11 5 (—1)"($ — 2)” 7) (8 points) The radius and interval of convergence of the power series 2 ( + 1)
n n=1 satisfy A) The radius is equal to 1 and the interval is (—1, 1). B) The radius is equal to 2 and the interval is (0,4).
C) The radius is equal to 1 and the interval is (1, 3).
D) The radius is equal to 1 and the interval is (1, 3]. E) The radius is equal to 1 and the interval is [1, 3]. 8) (8 points) Which of the following is a power series representation of the function
a: — 2
: ——————7
00 1 n
A) Z W — 2)
n:0 '
B) Z n (x — 2)
n=O
C) __ 2)n+1
71:00 Zh+ I
D) Z<~1>n<m — 2W
n20
0° (4)“
E 2 n+1
) ; (n+1)($ ) 6 1
(4 * SE)3
(Hint: Start with the power series of (4 — :10)"1 and differentiate it enough times.) 9)(8 points) The Maclaurin series of the functon f = is 71:2 2W“)
B) {rm—2
n=2
C) i (0"???  1) [En—2
D) :55 (4)2237: — 1) $712
71:2 10) (8 points) The Maclaurin series of f = (cos 1:)2 is equal to
1
(Hint: Use that (cos 302 = —(1 + cos 2
A) —21—+ ; (711,)” 51:”
B) :— + g :62"
0> %+ SEES?” D) ';‘+ S6221?!” 96”
E) g + f: (nll)" m4” 11)(8 points) Let = f: — 2)". We can say that the ﬁfth derivative of f at the point 2 is equal to n20 . A) f<5>(2) = 10. B) f<5>(2) = 64. C) f<5>(2) = 32. D) f<5>(2) = 21. E) f<5>(2) = 100. 12) (8 points) If we use that 11_ m = 1 + i m", and that
n=l % arcsin x = Vii—W, we conclude that the Maclaurin series of arcsinx is equal to A> w+ ————1':,;(:;:i;; U 8 13) (8 points) Let f(x) be a function dened on [1, 00) such that f(1:) > 1 for all :1: and hm % = 1. What can we say about the convergence of the series
Sl=isin(—1—)and32=isin< 1 >?
M f (n) f (n)3 A) 31 and 32 diverge. B) 31 converges and 5'2 diverges.
C) 5'1 diverges and 32 converges.
D) 31 and 32 converge. E) Nothing can be said about the convergence of the series. ...
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 Fall '11
 PETERCOOK
 Geometry, Mathematical Series, Scantron Sheet

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