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rn+1 1 rn+1 1 dt t 1 - dt > t 1 a) Show that In(n + 1) – In n = || - dt - t n+ 1° n

rn+1 1 rn+1 1 dt t 1 - dt > t 1 a) Show that In(n + 1) – In n = || - dt - t n+ 1° n

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.7: More On Inequalities
Problem 44E
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rn+1 1 rn+1 1 dt = t 1 dt > t 1 4. (a) Show that In(n + 1) – Inn = J1 t n + 1 n 1 1+ 1 1 (b) Let an +...+ 3 - In n, for n = 1,2, .... Use (a) to show that {an}-1 is a decreasing sequence. (Hint: Show that an an+1 > 0.) pn+1 1 (c) Use the left sum of 1 1 dt with partition {1,2, ..., n+1} to show that 1+ + +- > t In(n + 1). (d) Use (c) and the definition of an to show that an > 0 for all n > 1. (e) Use (b) and (d) to show that {an}1 converges to a number r. (r 0.577216, and is known as the Euler-Mascheroni constant.)
Transcribed Image Text:rn+1 1 rn+1 1 dt = t 1 dt > t 1 4. (a) Show that In(n + 1) – Inn = J1 t n + 1 n 1 1+ 1 1 (b) Let an +...+ 3 - In n, for n = 1,2, .... Use (a) to show that {an}-1 is a decreasing sequence. (Hint: Show that an an+1 > 0.) pn+1 1 (c) Use the left sum of 1 1 dt with partition {1,2, ..., n+1} to show that 1+ + +- > t In(n + 1). (d) Use (c) and the definition of an to show that an > 0 for all n > 1. (e) Use (b) and (d) to show that {an}1 converges to a number r. (r 0.577216, and is known as the Euler-Mascheroni constant.)
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