MATH 201 Exam 2, Fall 2015

Fall 2015-2016 Quiz 1 Prof. M. Egeileh Math 201 - Sections 29 to 32 Date: Oct Problem 1 (answer on pages 1 and 2 of the bi Which of the following series converge, and which divergs series. (8 pts each) oo font (—1)"° wae cos(1 + \ n — 5 aay < a) 5n-20 Jnl 1 aos 7 L4 fn | ae i n=3 n=2 Problem 2 (answer on pages 3 and 4 of the booklet Find the interval of convergence of the power series : == — Problem 3 (answer on pages 5 and 6 of the booklet 1. Prove that eae me ae De n! n=0 for all x € R. (9 pts) 2. Use the alternating series estimation theorem to prove that e < 3 (Hint : estimate 5 l i. s = ae the error resulting from the approximation e 7. 3 — (7 nis k=0 1 3. Let f(x) = a(& —e-*). Use the result of part 1. to find a power series expansion of f(z about the point x = 0. (6 pts) 4. Find the Taylor polynomial p4(z) generated by f(x) at x = 0. Then use Taylor’s theo- rem to estimate the error resulting from the approximation f(1) ~ ps(1). Does this approximation constitute an under-estimate or an over-estimate? (9 pts) (Hint : check that t(e* +e") is increasing on (0,00) and then use the result of part 2.) Problem 4 (answer on last page of the booklet) Let f(x) bea differentiable function whose derivative f ‘(x) is positive, continuous and decrea- & sing. For n > 1, let an = f (n) and b, = f'(n). Finally, assume that the series we b, converges. n=1 (A) Show that any1— On < bn for all n > 1. (5 pts) ~ / 2. Use the result of part 1. to prove that the sequence {a,} converges, and that Y = Via a, + Don > lim an. (7 pts) n=1 2 Prove that Dos — & — a; + lim a,. (5 pts) n->0o