School of Mathematics & Applied Statistics MATH188: Algebra and Differential Calculus Tutorial Assignment 2 – Spring 2020 Student Name: Student Number: This assignment is to be submitted via moodle by 11.55pm on Thursday 17th September (Week 7). Full working is to be shown for all solutions. All notation is as given in lectures. Untidy or badly set out work will not be marked and will be recorded as unsatisfactory, as will any question that uses the ∴ symbol or ⇒. If you use a result we have proved, please cite the result from the notes. Completed assignments must be submitted as a single pdf file through the moodle assignment submission portal. The assignment sheet will be the cover sheet. Note that it is good practice to number the individual pages of your assignment. This assignment is meant to be done individually. Cheating or plagiarism may lead to expulsion. Make sure to keep a copy of your assignment. Declaration: This assignment is all my own work and is not shared with or copied from any other person. Signature: Date: 1. For what values of a ∈ R does the series ∞∑ n=1 ( −4 2 + a )n converge? 2. Let a > 1. (a) Show that an > n(a− 1) for all n ≥ 1. [Hint: Write a = 1 + b, where b > 0, and use the Binomial Theorem. ] (b) Use the above to prove that lim n→∞ 1 an = 0. (c) Deduce using l’Hoˆpital’s rule that lim n→∞ n an = 0. (Note: by the definition of a sequence, you cannot differentiate a sequence.) (d) Use (c) and the Comparison Ratio Test to show that ∞∑ n=1 1 en − n converges. 3. Determine, justifying your conclusions, if the following series converge absolutely, converge conditionally or diverge. (a) ∞∑ n=1 (−1)nn6 3n6 − n (b) ∞∑ n=1 n3 + sinn 2n5 − n3 (c) ∞∑ n=1 n! (2n)! (d) ∞∑ n=1 (−1)n ln (n) n (e) ∞∑ n=1 (−1)nn2 2n 4. Write down the form of the partial fraction decomposition for the following rational functions. Do not find the numerical values of the constants. (a) x3 + 4 (x + 3)(x− 1)(x2 − 3) (b) 4×2 + 1 x3(x + 1)(x2 − 1)(x2 + 2x + 2)2 5. Evaluate the following integrals. (a) ∫ sin 4x cos 3x dx (b) ∫ sin3 x cos4 x dx (c) ∫ x2e−x dx (d) ∫ x2 dx x2 + 4x (e) ∫ dx x2 + 4x + 4 (f) ∫ dx x2 + 4x + 5 (g) ∫ √ 9 + x2 dx (h) ∫ ∞ 0 dx x2 + 9