Unformatted text preview: graph at the point with polar coordinates (r, θ) =
(1, π/6).
c) Express as a deﬁnite integral the area of the part of the region in a) that lies outside the circle
of radius 1 centered at the origin. Do not evaluate the integral.
2.17 Find the limits of each of the following sequences as n → ∞ or say that the limit does
not exist.
4n2 + 3n
a) sn =
(1 + n)(1 + 2n)
(−1)n
b) sn = 3 + √
n
n2
c) sn = n
2 MTH 142 Spring 2011 Exam 2 Practice 4 2.18 a) Find the sum of the inﬁnite series
3 − 3/2 + 3/4 − 3/8 + 3/16 − 3/32 + 3/64 − 3/128 + · · ·
.
b) Find an explicit formula in terms of a for the sum
21 3a2n .
n=1 MTH 142 Exam 3 Practice Problems Spring 2011 1 No calculators will be permitted at the exam.
3.1 A pingpong ball is launched straight up, rises to a height of 15 feet, then
falls back to the launch point and bounces straight up again. It continues to
bounce, each time reaching a height 90% of the height reached on the previous
bounce. Find the total distance that the ball travels.
Note: Questions about convergence or divergence of series may be posed
in various ways – some purely multiple choice, others requiring a coherent
(speciﬁc or nonspeciﬁc) reason, some requiring choice and supporting detail.
Read questions carefully!
3.2 Use the integral test to determine the convergence of the following series: b) ∞ 1 n=1 a) n 3/2 ∞ n
n
n=1 e 3.3 Determine if the following series converge or diverge. Give your reasoning
using complete sentences.
a)
b) ∞ ln n
2
n=1 n
∞ n!
n=1 (n + 2)! 3.4 For each of the following items a) and b) choose a correct conclusion
and reason from among the choices (R), (C), (I) below and provide supporting
computation. For example, if you choose (R) calculate and interpret a suitable
ratio. MTH 142 Exam 3 Practice Problems Spring 2011
(R) Converges by the ratio test. 2 (C) Diverges by a pseries comparison. (I) Converges by the integral test. a)
b) ∞ n2 3n
n
n=1 n4
∞ ln n
n=1 n
∞ 3n + 4n
and
en
n=1
gives a valid reason? Circle your answer. No other reas...
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 Fall '08
 POOLE
 Calculus, Angles, Mathematical Series, Practice Problems Spring

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