Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Find two or three classmates and a few feet of chalkboard As a group try your hand at the following exercises Be sure to discuss how to solve the exercises how you get the solution is much more important than whether you get the solution If as a group you agree that you all understand a certain type of exercise move on to later problems You are not expected to solve all the exercises some are very hard Exercises marked with an are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart Others are my own or are independently marked Infinite Series A series is an infinite sum of numbers There are two sequences associated to each series First of all there s the sequence of terms in the sum a k a summands 1 1 1 1 2 4 8 1 1 1 1 2 4 8 Second there s the sequence of partial sums 1 1 1 1 2 4 8 3 7 15 1 2 4 8 We normally write the former explicitly X 1 1 1 1 1 2 4 8 2n n 0 On the other hand a series converges exactly if the sequence of partial sums converges and the limit of the sequence of partial sums is the value of the series Pn Standard notation the sequence of summands is a the sequence of partial sums is s n n k 0 ak and if sn converges then P a lim s n n n 0 n Here are some facts about series A series cannot converge unless the sequence of summands tends to 0 But the sequence of summands can go to 0 without the series converging P n The geometric series a ar ar2 ar3 n 0 ar converges if and only if r 1 If it converges then it converges to a 1 r A proof and generalization of this is in exercise 3 P P P Pn If a A and b B then a b A B Let c n n n n n k 0 ak bn k Then P cn AB This is called the Cauchy product or discrete convolution of the two series This one isn t so much a fact as a technique P If we can write each an as a difference an bn bn 1 then sn b0 bn 1 so the sum an converges if the sequence bn converges Partial fractions are a good way to find such decompositions of terms as differences 1 What is the sum n X 1 n 1 2 1 1 1 1 2 4 8 16 1 How about n X 1 n 1 3 1 1 1 1 3 9 27 81 If k is some number strictly greater than 1 what is n X 1 1 1 1 1 2 3 4 k k k k k n 1 2 Determine whether the following series are convergent or divergent If convergent find the sums a d g X 3 n 1 n 1 X n 0 X n 0 4n 3n b 1 2n e 3 n n 3 h X 1 n 2 n 0 X n 0 X c 8 n 3 n f arctan n i n 1 X n 3n 1 n 1 X cos 1 n n 0 X n 1 ln n n 1 3 Remember how to evaluate the geometric series if r 1 then we can evaluate S P n a ar ar 2 ar 3 by mutiplying by r and subtracting ar n 0 S a ar ar2 ar3 r S ar ar2 ar3 S rS a a thus S 1 r Use this method to compute the following sums a 1 2 3 4 5 6 3 9 27 81 243 b 1 4 9 16 25 36 2 4 8 32 64 4 When money is spent on goods and services those who receive the money also spend some of it The people receiving some of the twice spent money will spend some of that and so on this chain reaction is called the multiplier effect In a hypothetical isolated community the local government begins the process by spending D dollars Suppose that each recipient of spent money spends 100c and saves 100s of the money she or he receives The values c and s are called the marginal propensity to consume and the marginal propensity to save and of course c s 1 a Let Sn be the total spending that has been generated after n transactions Find an equation for Sn b Show that limn Sn kD there k 1 s is the multiplier What is the multiplier if the marginal propensity to consume is 80 c In fact the marginal propensities to save and to consume vary by socioeconomic class among other things poor people spend a larger proportion of the money they receive and rich people save a larger proportion If the government is trying to stimulate the economy to whom should it give its money 2
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