MATH 201 Exam 1, Spring 2017

Math 201 - Quiz 1 (Spring 17) T. Tlas ================================================ • Write the answers to questions 2, 3 and 4 on their sheets. The first question has an extra sheet for you to write your answers on it. Any part of your answers written on the wrong sheet will not be graded. Note that a sheet of paper has two sides. • There are 4 problems in total. Some questions have several parts to them. Make sure that you attempt them all. • This is a closed book exam and no calculators are allowed. ================================================ Name : ID # : Section : ================================================ Q1 Q2 Q3 Q4 TOTAL Problem 1 (14 points each) Which of the following series converge and which diverge? Those which converge, do they converge absolutely or conditionally? When possible find the sum of the series. i- ∞ � n=1 (−1)n ln(n) n 5 4 ii- ∞ � n=0 (−1)n π2n+1 (2n + 1)! 22n+1 iii- ∞ � n=1 � 1 − 2 5n �n iv- ∞ � n=1 n2� cos � 1 n � − 1 + 1 2n2 � ================================================ ADDITIONAL SHEET FOR PROBLEM 1 ANSWER Problem 2 (17 points) Evaluate the following integral � 1 0 sin(x3)dx with an error less than 0.001. Is your answer an over- or an under-estimate? ================================================ Problem 3 (17 points) Find the Taylor series at 0 of e−x3. What is its radius of convergence? Estimate e−0.008 with an error less than 10−6. Is your answer an under- or an over-estimate? ================================================ Problem 4 (10 points) Suppose �∞ n=1 an is a series such that an ≥ 0 and an ≥ an+1 for all n. If you know that the series ∞ � n=0 2n a2n = a1 + 2a2 + 4a4 + 8a8 + . . . converges, does it follow that the original series, i.e. ∞ � n=1 an = a1 + a2 + a3 + a4 + . . . converges as well? ================================================