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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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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answer the #55 thank you
54. If s, is the kth partial sum of the harmonic series. prove that In(k + 1) < Sk < 1 + In k. Hint: 1 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 + + ... + 51 1 Σ m=50 m 50 100
Transcribed Image Text:54. If s, is the kth partial sum of the harmonic series. prove that In(k + 1) < Sk < 1 + In k. Hint: 1 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 + + ... + 51 1 Σ m=50 m 50 100
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