Description

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 (8pts) [Ste10, p. 720] Use integral test to determine whether the series is convergent or divergent

Exercise 7.2 (2pts) [Ste10, p. 727] For what values of p ∈ R does the series converge?
Exercise 7.3 (2pts) [Ste10, p. 727] Show that if a ≥ 0 and Xan < ∞, then .
Exercise 7.4 Work out the details of using Shanks transformation to calculate S◦3(S3) of the series

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

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

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

Exercise 7.8 (8pts) [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) = x2 arctan(x3)
Exercise 7.9 (8pts) [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 3 −2. (ii) f(x) = 1/x, a = −3. (iii) f(x) = sinx, a = π/2. (iv) f(x) = √x, a = 16. − x , a =
Exercise 7.10 (4pts) Find general solution x(t) to the following ODE’s
(i) x¨ + 4x˙ + 5x = e5t + te−2t cost (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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