54. If s, is the kth partial sum of the harmonic series. prove that In(k + 1) < S < 1 + In k. Hint: 1 1 if 0 < m < x Sm + 1. Integrate - т+1 each member of the inequality from m to m + 1; let m take on successively the values 1, 2, . . . , n - 1, and add the results. m 55. Use the method of Exercise 54 to estimate the sum 100 1 1 50 + ... + 100 m=50 m 51

54. If s, is the kth partial sum of the harmonic series. prove that In(k + 1) < S < 1 + In k. Hint: 1 1 if 0 < m < x Sm + 1. Integrate - т+1 each member of the inequality from m to m + 1; let m take on successively the values 1, 2, . . . , n - 1, and add the results. m 55. Use the method of Exercise 54 to estimate the sum 100 1 1 50 + ... + 100 m=50 m 51

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 68E

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