Math 1b Midterm II Review T Judson April 14 2006 1 Resources for Review Midterm II Review Guide http www courses fas har vard edu math1b exams Midterm2Sp05review pdf Exams and solutions from previous years http www courses fas har vard edu math1b prevexams Solutions to the Chapter 8 Review Exercises http www courses fas har vard edu math1a exams 2 Exam Particulars Tuesday April 18 at 7 9 PM in Science Center B No calculators allowed All out of sequence exams must be approved by the course head No make up exams 3 What to Expect Approximately nine questions some with several parts The emphasis will be on material from Chapter 8 omit Section 8 8 Refer to the Midterm II Review Guide for details 4 Sequences A sequence is a list of numbers a1 a2 a3 More formally we can think of a sequence as a function a N R where a n an Give a example of a sequence that is 1 convergent but not monotone 2 monotone but not convergent 3 bounded but not monotone 4 monotone decreasing and unbounded 5 monotone increasing and convergent 6 unbounded but not monotone 7 bounded and monotone but not convergent 5 Geometric Series Let k 0 ar k a ar ar 2 ar 3 If r 1 then k 0 ar k a 1 r If r 1 the series diverges 1 Express 0 131313 as a ratio of two integers 2 A ball is dropped from a height of 8 ft Each time the ball bounces it comes back up to one half of its previous height What is the total distance that the ball travels 6 Series n 1 2 n 2 n n 1 2 2 n 1 n 1 3 n2 2n 5n n 2 n 4 k 1 2k 3k 4 k ln n 5 2 ln 3 n n 1 6 n2 ne n 1 7 1 If the series is of the form 1 np then it is a pseries The series converges for p 1 and diverges for p 1 2 If the series has the form ar n then it is a geometric series and converges for r 1 and diverges for r 1 3 If the series is similar to a p series or a geometric series consider the Comparison Test 4 If limn an 0 then the series diverges 5 If the series is of the form 1 n 1 an consider applying the Alternating Series Test You can also test for absolute convergence 6 If the series involves products factorials or constants raised to the nth power consider the Ratio Test 7 If an f n and the integral 1 f x dx is easily evaluated the Integral Test may be useful assuming the hypothesis of the test are satisfied 8 Is the series a telescopic series If so convergence or divergence can be determined by computing the limit of the partial sums of the series 2 8 Alternating Series Let n 1 1 n 1an a1 a2 a3 a4 satisfy the following conditions 1 a1 a2 a3 2 lim an 0 n Then the series converges and Rn S Sn an 1 9 Power Series Representations Find the interval of convergence of the power series x 1 k k k4 k 1 10 Important Power Series Function ex sin x cos x 1 1 x Series x2 x3 1 x 2 3 3 5 x x x 3 5 2 x4 x 1 2 4 1 x x2 x 3 Interval of Convergence 1 1 3 11 Taylor Series Find the Taylor series centered at x 0 for ex e x sinh x 2 12 Taylor Polynomials Find the second degree Taylor polynomial for f x x at the number a 100 Approximate 101 Estimate the error between the approxima tion and 101 4 13 Taylor Polynomials and Error If f n 1 M between a and x then the remainder of the Taylor series satisfies the inequality M f x Tn x Rn x x a n 1 n 1 14