The harmonic series and Euler's constanta. Sketch the

Chapter 10, Problem 62AE

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The harmonic series and Euler's constant

a. Sketch the function f(x) = 1/x on the interval [1, n + 1], where n is a positive integer. Use this graph to verify that

           \(\ln (n+1)<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}<1+\ln n\)

b. Let \(S_{n}\) be the sum of the first n terms of the harmonic series, so part (a) says \(\ln (n+1)<S_{n}<1+\ln n\). Define the new sequence \(\left\{E_{n}\right\}\), where

                       \(E_n=S_n-\ln(n+1)\),     for n = 1, 2, 3, . . . .

Show that \(E_{n}>0\) for n = 1, 2, 3, . . . .

c. Using a figure similar to that used in part (a), show that

               \(\frac{1}{n+1}>\ln (n+2)-\ln (n+1)\)

d. Use parts (a) and (c) to show that \(\left\{E_{n}\right\}\) is an increasing sequence \(\left(E_{n+1}>E_{n}\right)\).

e. Use part (a) to show that \(\left\{E_{n}\right\}\) is bounded above by 1.

f. Conclude from parts (d) and (e) that \(\left\{E_{n}\right\}\) has a limit less than or equal to 1. This limit is known as Euler's constant and is denoted \(\gamma\) (the Greek lowercase letter gamma).

g. By computing terms of \(\left\{E_{n}\right\}\), estimate the value of \(\gamma\) and compare it to the value \(\gamma \approx 0.5772\). (It has been conjectured, but not proved, that \(\gamma\) is irrational.)

h. The preceding arguments show that the sum of the first n terms of the harmonic series satisfy \(S_{n} \approx 0.5772+\ln (n+1)\). How many terms must be summed for the sum to exceed 10?