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3.5 a) Which of the following correctly classiﬁes the series i) Diverges: limn→∞ an = 0.
ii) Converges: sum of two convergent geometric series.
iii) Converges: Comparison test with geometric series with ratio 3/e.
iv) Converges: Comparison test with geometric series with ratio e/4.
v) Diverges: constant multiple of the harmonic series.
b) Which of the following correctly classiﬁes the series
and gives a
n=1 5 + n
valid reason? Circle your answer. No other reason required.
i) Diverges: limn→∞ an = 0.
ii) Converges: Ratio test.
iii) Converges: Comparison test with a p-series, p > 1.
iv) Diverges: Ratio test.
v) Diverges: Comparison or limit comparison with the harmonic series.
3.6 Determine if the following series converge or diverge. Give your reasoning
using complete sentences. MTH 142 Exam 3 Practice Problems Spring 2011 a) ∞
n=1 b) (−1)n ∞
n=2 3 n
n+2 (−1)n+1 1
ln n 3.7 a) Find the interval of convergence for the power series
b) Find the radius of convergence for ∞ 1
(x − 3)n
n=1 n2 ∞ n! n
n=1 (2n)! 3.8 a) Find the Taylor series about x = π of cos(x). Your answer should clearly
indicate the pattern of the terms. (Note the center, π .)
b) Find the Taylor series of ln(x) at x = 1.
3.9 Give the Taylor polynomial at 0 of degree 4 for each of the following functions. Use the easiest method you know, including your knowledge of the Taylor
series for sinx, ex , (1 + x)p , etc.
a) f (x) = x2 cos(x).
b) g (x) = √
c) f (x) = 1
1 − 2x x
0 sin(t2 ) dt d) f (x) = (1 + x) sin(2x)
3.10 The graph of y = f (x) passes through the point (0, −2) and near that
point the function is decreasing and concave up. If the third degree Taylor
polynomial of f near 0 is P2 (x) = a + bx + cx2 what are the signs of a, b and c?
2 b) If f (x) = e−x ﬁnd f (8) (0) and f (9) (0), that is, the 8th and 9th derivative of
f at 0. Suggestion: Consider the Taylor series of f and look at the coeﬃcients
of x8 and x9 ....
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