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 Harmonic SeriesThe harmonic series is the sum∞Xn=11n= 1 +12+13+14+15+ . . .The sum diverges, although it does so slowly. A classic proof that the harmonic series diverges is thefact that13+14>12,15+16+17+18>12, etc., so the sum is at least 1+12+12+12+· · · = 1+12×∞ = ∞.The harmonic series can be factored:1 +12+13+14+15+ · · · =1 +12+14+18+ . . .1 +13+15+17+ . . .The first term is justP∞012n= 1/(1 −12) = 2. The second term can be factored again:1 +13+15+17+19+ · · · =1 +13+19+127+ . . .1 +15+17+111+ . . .and repeated factorization gives:∞ =∞Xn=11n=1 +12+14+ . . .1 +13+19+ . . .1 +15+125+ . . .. . .=213254· · · =Yprimes ppp − 1Each term on the right-hand-side is summed using the geometric series. A prime is a positivenumber p with precisely two factors. If there were only finitely many primes, then the product onthe right-hand-side would be a finite number. Thus, there must be infinitely many primes.1. The following steps will give another proof that the harmonic series diverges:(a) Consider the alternating harmonic series:1 −12+13−14+15−16+ · · · =∞Xn=1(−1)n−11nLet A represent the alternating harmonic series and H the harmonic series. Explain whyA = H − H.1(b) We will prove later that the alternating harmonic series converges, to some positivenumber. (In fact, A = ln 2.) But if H were convergent, what must A converge to?2. It’s a fact, although rather difficult to prove, that∞Xn=11n2= 1 +14+19+125+136+ . . .converges to π2/6. (We will prove that it converges tomorrow.) Find a factorization of theabove sum analogous to the factorization of the harmonic series. Use the fact that π2isirrational to prove that there are infinitely many prime numbers.Series are a lot like integrals. If anbe a sequence (starting at 0, say), let’s define its discreteintegral to be the sequence of partial sums, given by:(Sa)n=n−1Xk=0akfor n ≥ 1 and by (Sa)0= 0. Also, we define the discrete derivative of anto be the sequence:(Da)n= an+1− anIt’s now straightforward to check the discrete fundamental theorem of calculus:(DSa)n= anand (SDa)n= an− a0Then the infinite seriesP∞n=0anis just limn→∞(Sa)n.3. Compare the definition of the infinite seriesP∞n=0anto the definition of the improper integralR∞0f(x)dx.4. Find a formula for the “discrete product rule”: I.e. find a formula for (D(ab))n, where anandbnare sequences and the product sequence (ab)nis defined to be (ab)n= anbn.5. Use your answer to the previous question to find a formula for the “discrete integration
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