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Vv156 Honors Calculus II  Assignment 7

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Vv156 Honors Calculus II
Assignment 7

This assignment has a total of (54 points).
Note: Unless specified otherwise, you must show the details of your work via logical reasoning for each exercise.
Simply writing a final result (whether correct or not) will receive 0 point.
Exercise 7.1 (8 pts) [Ste10, p. 720] Use integral test to determine whether the series is convergent or divergent
X∞
n=1
1
√5 n
(i) X∞
n=1
1
(2n + 1)3
(ii) X∞
n=1
n
n2 + 1
(iii) X∞
n=1
n
2
e
−n
3
(iv)
Exercise 7.2 (2 pts) [Ste10, p. 727] For what values of p ∈ R does the series X∞
n=2
1
np ln n
converge?
Exercise 7.3 (2 pts) [Ste10, p. 727] Show that if a ≥ 0 and Xan < ∞, then Xa
2
n < ∞.
Exercise 7.4 Work out the details of using Shanks transformation to calculate S ◦3
(S3) of the series
X∞
k=0
(−1)k
2k + 1
= 1 −
1
3
+
1
5

1
7
+ · · ·
Exercise 7.5 (8 pts) [Ste10, p. 737] Determine whether the series is absolutely convergent, conditionally convergent,
or divergent.
X∞
n=1
n
5
n
(i) X∞
n=1
(−1)n−1 n
n2 + 4
(ii) X∞
n=2
(−1)n
ln n
(iii) X∞
n=1
(2n)!
(n!)2
(iv)
Exercise 7.6 (8 pts) [Ste10, p. 745]
X∞
n=1
(−1)nnxn
(i) X∞
n=1
(−x)
n
n2
(ii) X∞
n=2
(−x)
n
4
n ln n
(iii) X∞
n=2
x
2n
n(ln n)
2
(iv)
Exercise 7.7 (4 pts) [Ste10, p. 751] Express the function as the sum of a power series by first using partial fractions.
Find the interval of convergence.
f(x) = 3
x
2 − x − 2
(i) f(x) = x + 2
2x
2 − x − 1
(ii)
Exercise 7.8 (8 pts) [Ste10, p. 752] Find a power series representation for the function and determine the radius of
convergence.
(i) f(x) = ln(5 − x) f(x) = x
2
arctan(x
3
(ii) ) f(x) = x
(1 + 4x)
2
(iii) f(x) = x
2 − x
(1 − x)
3
(iv)
Exercise 7.9 (8 pts) [Ste10, p. 765] Find the Taylor series for f(x) centered at the given value of a. [Assume that f
has a power series expansion.] Also find the associated radius of convergence.
f(x) = x − x
3
(i) , a = −2. (ii) f(x) = 1/x, a = −3. (iii) f(x) = sin x, a = π/2. f(x) = √
(iv) x, a = 16.
Exercise 7.10 (4 pts) Find general solution x(t) to the following ODE’s
x¨ + 4 ˙x + 5x = e
5t + te−2t
(i) cost x¨ + 4 ˙x + 4x = t
2
e
−2t
(ii)
References
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on page 1).
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