VV156 Homework 7 Solved

This assignment has a total of
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 ∞∞∞∞

􏰄 1 (i) √5 n

(ii)

􏰄 1

(iii)

􏰄 n 􏰄 2 −n3 n2 +1 (iv) n e

(2n+1)3
Exercise 7.2 (2 pts) [Ste10, p. 727] For what values of p ∈ R does the series 􏰄∞ 1 converge?

np ln n

n=2 Exercise7.3 (2pts)[Ste10,p.727]Showthatifa≥0and􏰄an <∞,then􏰄a2n <∞.

Exercise 7.4 Work out the details of using Shanks transformation to calculate S ◦3(S3) of the series 􏰄∞ (−1)k 1 1 1

Exercise 7.5 (8 pts) [Ste10, p. 737] Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

n=1

n=1

n=1 n=1

(ii) Exercise 7.6 (8 pts) [Ste10, p. 745]

(iii)

lnn

(iv)

(n!)2

x2n n(lnn)2

(−1) n=1

2k + 1 = 1 − 3 + 5 − 7 + · · · k=0

∞∞∞∞

􏰄 n (i) 5n

􏰄

n−1

n n2 +4

􏰄 (−1)n

􏰄 (2n)!

(i)

􏰄 n (−1) nx

n

(ii)

􏰄 (−x)n

(iii)

􏰄 (−x)n

(iv)

􏰄

n=1

n=2

n=1

∞∞∞∞

n2
Exercise 7.7 (4 pts) [Ste10, p. 751] Express the function as the sum of a power series by first using partial fractions.

n=1

n=1

n=2

4nlnn

n=2

Find the interval of convergence.
(i) f(x) = 3 (ii) f(x) = x + 2

x2 −x−2 2×2 −x−1
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) (ii)f(x)=x2arctan(x3) (iii)f(x)= x (iv)f(x)= x2−x

(1+4x)2 (1−x)3 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.
(i) f(x) = x − x3, a = −2. (ii) f(x) = 1/x, a = −3. (iii) f(x) = sin x, a = π/2. (iv) f(x) = √x, a = 16.

Exercise 7.10 (4 pts) Find general solution x(t) to the following ODE’s
(i) x ̈ + 4x ̇ + 5x = e5t + te−2t cos t (ii) x ̈ + 4x ̇ + 4x = t2e−2t

References

[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on page 1).

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