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 The Integral Test LetR f x be a continuous function that P is positive and decreasing at least after some cut off Then 0 f x dx converges if and only if 0 f n converges Indeed we have a much more precise theorem Let f x be a positive decreasing function on N where N is an integer Then Z X f x dx N 1 Z f n f x dx N n N 1 P then we define the N th remainder to be If the infinite sum 1 an converges to some number s PN P 1 an The N th remainder measures the N 1 an s sN where sN is the N th partial sum error in estimating the infinite sum by the N th partial sum Thus the integral test provides a bound on the error of the estimate important application of the integralR test is to determine whether the p series P A particularly p p 1 1 n converges Recall that the improper integral 1 1 x dx converges if and only if p 1 then the infinite sum follows the same rule 1 Determine whether the following series converge or diverge a d X 1 X 1 g ne n b 3n 2 n n 1 e 1 X 1 1 n X e 1 X n 2 h n2 n 1 c 1 n2 4n 5 X n2 3 1 f X 2 i en X X 1 n2 n3 1 1 n ln n 2 n n4 1 1 np converge P b For what values of p does 1 n ln n p converge You may assume that the series starts after n 1 2 a For what values of p does P c For what pairs of values p0 p1 does X np 0 1 ln n p1 converge You may assume that the series starts after n 1 1 d For what k 1 tuples p0 p1 pk does X 1 ln n pk np0 ln n p1 ln ln n p2 ln z k converge You may assume that the series starts late enough so as never to have 0s in the denominator P 3 Let sn nk 1 k1 be the nth partial sum of the harmonic series a Draw a picture to prove that ln n sn 1 ln n b Draw a picture to determine whether the sequence sn ln n is increasing decreasing or not monotonic c Does the sequence sn ln n have a limit How do you know 4 Find all values of c for which the following series converges X c n 1 5 It is a fact that P n 1 1 n 2 1 n n 1 2 6 a Let s say you were to estimate the value of 2 6 by summing the first ten terms of the above infinite series How accurate is this estimate b How many terms would you need to sum to calculate 2 6 correct to ten decimal places 6 The integral test provide both upper and lower bounds for the sizes of errors in estimating series In this exercise we will describe a better method for estimating series a P Let f x be a positive decreasing function on 1 and an f n Assume that 1 an converges Prove that the number X N X an n 1 Z an n 1 f x dx N 1 is positive b By drawing a picture show that the above number is less than aN 1 f N 1 c Let s now assume that in addition to being decreasing and positive f x is also concaveup everywhere Prove that Z N X X 1 aN 1 an an f x dx aN 1 2 N 1 n 1 n 1 7 Use the results from the previous exercise to estimate 2 6 correct to ten decimal places 2
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