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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