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 Harmonic Series The harmonic series is the sum X 1 1 1 1 1 1 n 2 3 4 5 n 1 The sum diverges although it does so slowly A classic proof that the harmonic series diverges is the fact that 13 14 12 51 16 17 81 12 etc so the sum is at least 1 12 12 12 1 21 The harmonic series can be factored 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 2 4 8 3 5 7 P 1 n The first term is just 1 1 21 2 The second term can be factored again 0 2 1 1 1 1 1 1 1 1 1 1 1 1 1 3 5 7 9 3 9 27 5 7 11 and repeated factorization gives X 1 1 1 1 1 1 1 1 1 1 n 2 4 3 9 5 25 n 1 Y 2 3 5 p 1 2 4 p 1 primes p Each term on the right hand side is summed using the geometric series A prime is a positive number p with precisely two factors If there were only finitely many primes then the product on the 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 X 1 1 1 1 1 1 1 n 1 2 3 4 5 6 n n 1 Let A represent the alternating harmonic series and H the harmonic series Explain why A H H 1 b We will prove later that the alternating harmonic series converges to some positive number 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 X 1 1 1 1 1 1 2 n 4 9 25 36 n 1 converges to 2 6 We will prove that it converges tomorrow Find a factorization of the above sum analogous to the factorization of the harmonic series Use the fact that 2 is irrational to prove that there are infinitely many prime numbers Series are a lot like integrals If an be a sequence starting at 0 say let s define its discrete integral to be the sequence of partial sums given by Sa n n 1 X ak k 0 for n 1 and by Sa 0 0 Also we define the discrete derivative of an to be the sequence Da n an 1 an It s now straightforward to check the discrete fundamental theorem of calculus DSa n an and SDa n an a0 Then the infinite series P n 0 an is just limn Sa n 3 Compare the definition of the infinite series R f x dx 0 P n 0 an to the definition of the improper integral 4 Find a formula for the discrete product rule I e find a formula for D ab n where an and bn are sequences and the product sequence ab n is defined to be ab n an bn 5 Use your answer to the previous question to find a formula for the discrete integration by parts 2
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