Math 1b Final Review Series T Judson May 17 2006 Resources for Review Review Guides http www courses fas har vard edu math1b exams Exams and solutions from previous years http www courses fas har vard edu math1b prevexams Solutions to the Chapter Review Exercises http www courses fas har vard edu math1a exams Exam Particulars Tuesday May 23 at 2 15 5 15 PM in Geology Lecture Hall No calculators allowed All out of sequence exams must be approved by the Final Exams Office Geometric Series Let k 0 ar k a ar ar 2 ar 3 If r 1 then k 0 a ar 1 r k If r 1 the series diverges Suppose that 100 is initially deposited in a bank Experience has shown bankers that that on the average only 8 of the money deposited is withdrawn by the owner If r 1 then a ar 1 r k An Application k 0 If r 1 the series diverges Suppose that 100 is initially deposited in a bank Experience has shown bankers that that on the average only 8 of the money deposited is withdrawn by the owner at any time Consequently bankers feel free to lend out 92 of their deposits Thus 92 of the original 100 is loaned out to other customers to start a business for example This 92 will become someone else s income and sooner or later will be redeposited in the bank Then 92 of the 92 or 84 64 is loaned out and eventually redeposited Of the 84 64 the bank loans out 92 and so on What is the total amount of money deposited in the bank as a result of these transactions Series 1 4 n 1 n n 1 2 1 1 n n 2 3 n 0 sin 1 n 3 n n 1 2n 3 n ln n n 2 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 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 Absolute and Conditional n 2 Convergence3 n ln n n 2 1 n n ln n n 2 Power Series Representations Find the interval of convergence of the power series 1 k k 1 k 1 k x 4 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 Taylor Series 3 x e a Write down a power series expansion for b Write down a power series expansion for and determine its radius of convergence x3 e dx c Use your answer in part b to find a series for 1 2 x3 e dx 0 d If you approximate the definite integral 0 1 2 x3 e dx by taking the partial sum consisting of the first four nonzero terms of the series that you obtained in part c what is the maximum error for your approximation Taylor Polynomials Let f be a function having derivatives of all orders for all real numbers The third degree Taylor polynomial for f about x 2 is given by 1 3 2 x 2 3 T3 x 2 x 2 8 12 a Find f 2 f 2 and f 2 b Determine whether f has a local minimum a local maximum or neither at x 2 Justify your answer c Use T3 x to find an approximation for f 0 d The fourth derivative of f satisfies the inequality 1 4 f x 4 for all x in the closed interval 2 0 Find an error bound on the approximation for f 0 that you found in part c