Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Find two or three classmates and a few feet of chalkboard. As a group, try your hand at thefollowing exercises. Be sure to discuss how to solve the exercises — how you get the solution ismuch more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: some are very hard.Exercises marked with an § are from Single Variable Calculus: Early Transcendentals for UCBerkeley by James Stewart. Others are my own or are independently marked.The Integral TestLet f(x) be a continuous function that is positive and decreasing, at least after some cut-off.ThenR∞0f(x) dx converges if and only ifP∞0f(n) converges. Indeed, we have a much more precisetheorem. Let f(x) be a positive decreasing function on [N, ∞), where N is an integer. Then:Z∞N+1f(x) dx ≤∞Xn=N+1f(n) ≤Z∞Nf(x) dxIf the infinite sumP∞1anconverges to some number s, then we define the Nth remainder to beP∞N+1an= s − sN, where sNis the Nth partial sumPN1an. The Nth remainder measures theerror in estimating the infinite sum by the Nth partial sum. Thus, the integral test provides abound on the error of the estimate.A particularly important application of the integral test is to determine whether the p-seriesP∞11/npconverges. Recall that the improper integralR∞11/xpdx 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)∞X1ne−n(b)∞X1n + 2n + 1(c)∞X1n2n3+ 1(d)∞X13n + 2n(n + 1)(e)∞X11n2− 4n + 5(f)∞X21n(ln n)2(g)∞X1e1/nn2(h)∞X3n2en(i)∞X1nn4+ 12. (a) For what values of p doesP1/npconverge?(b) For what values of p doesP1/(n (ln n)p) converge? You may assume that the seriesstarts after n = 1.(c) For what pairs of values (p0, p1) doesX1np0(ln n)p1converge? You may assume that the series starts after n = 1.1(d) For what (k + 1)-tuples (p0, p1, . . . , pk) doesX1np0(ln n)p1(ln ln n)p2. . . (ln . . . ln| {z }kn)pkconverge? You may assume that the series starts late enough so as never to have 0s inthe denominator.3. § Let sn=Pnk=11kbe 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:∞Xn=1cn−1n + 15. It is a fact thatP∞n=11/n2= π2/6.(a) Let’s say you were to estimate the value of π2/6 by summing the first ten terms of theabove infinite series. How accurate is this estimate?(b) How many terms would you need to sum to calculate π2/6 correct to ten decimal places?6. The integral test provide both upper and lower bounds for the sizes of errors in estimatingseries. In this exercise, we will describe a better method for estimating series.(a) Let f(x) be a positive decreasing function on [1, ∞), and an= f (n). Assume thatP∞1anconverges. Prove that the number ∞Xn=1an!− NXn=1an+Z∞N+1f(x) dx!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 concave-up everywhere. Prove that:12aN+1≤ ∞Xn=1an!− NXn=1an+Z∞N+1f(x) dx!≤ aN+17. Use the results from the previous exercise to estimate π2/6 correct to ten decimal
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