The harmonic series and Eulers constant a. Sketch the

Chapter 8, Problem 70

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The harmonic series and Eulers constant a. Sketch the function f 1x2 = 1>x on the interval 31, n + 14, where n is a positive integer. Use this graph to verify that ln 1n + 12 6 1 + 1 2 + 1 3 + g+ 1 n 6 1 + ln n. b. Let Sn be the sum of the first n terms of the harmonic series, so part (a) says ln 1n + 12 6 Sn 6 1 + ln n. Define the new sequence 5En6 by En = Sn - ln 1n + 12, for n = 1, 2, 3, c. Show that En 7 0, for n = 1, 2, 3, c. c. Using a figure similar to that used in part (a), show that 1 n + 1 7 ln 1n + 22 - ln 1n + 12. d. Use parts (a) and (c) to show that 5En6 is an increasing sequence 1En + 1 7 En2. e. Use part (a) to show that 5En6 is bounded above by 1. f. Conclude from parts (d) and (e) that 5En6 has a limit less than or equal to 1. This limit is known as Eulers constant and is denoted g (the Greek lowercase gamma). g. By computing terms of 5En6, estimate the value of g and compare it to the value g _ 0.5772. (It has been conjectured that g is irrational.) h. The preceding arguments show that the sum of the first n terms of the harmonic series satisfy Sn _ 0.5772 + ln 1n + 12. How many terms must be summed for the sum to exceed 10?