MATH 201 Exam 1, Spring 2010

Spring 2009-2010 Quiz 1 Date: March 12 Prof: M. Egeileh Math 201 - Sections 1 to 3 Duration: 1 hour Problem 1 (answer on pages 1 and 2 of the booklet) Which of the following series converge, and which diverge? When possible find the sum of the ah a series. (9 pts each) i i Vie: iy 3 (In n)?04 = ee 8\" “. sin(1 +n!) OGG) OLEH oper (+2) ape n=1 n=0 Problem 2 (answer on pages 3 and 4 of the booklet) n+l y) 7, Find the interval of convergence of the power series : Gann — 2)", n=1 For which values of x does the series converge absolutely ? Conditionally? (20 pts) Problem 3 (answer on pages 5 and 6 of the booklet) 1. Prove that ope n=0 nl for all x € R. (10 pts) 2. Use the alternating series estimation uheoven to estimate the error resulting from the =~ a ier oi approximation e7! ~ 9s, where 95 = a ( = Does this approximation tend to be too : k=0 ‘ small or too large? (7 pts) 1 3. Let f(z) = ria +e-*). Use part 1. to find a power series expansion of f(x) about x = 0. Ct “¢ Here, you ay find it easier to write series in expanded way, rather than use the sigma Lb notation. (4 pts) 7 5 4. Prove that for every c such that 0 <c < 1, we have f(c) < <3 (Hint : you may prove that } < In3 and use the fact that the ae f(z). is increasing for x > 0). (5 pts) Tae sil 2 : O07 eS 5. Find the Taylor polynomial p3(z) generated by f(z) at the point z = 0. Then use Taylor’s 2 theorem to estimate the error resulting from the approximation f(1) * ps(1). (6 pts) Problem 4 (answer on last page of the booklet and its back) Let {a,} be the sequence defined by : eas 2 8 Sor 1. Prove that In(n+1) <a, <1+In(n) for alln 21. (6 pts) z co 3 2. Does the sequence {a,} converge? What about the series y i e j 2 ZA n(n =2 A : Justify your answers. (6 pts) a a I