This project is based off of two separate results which show the utility of series and sequences.The first of these is about In(2).The goal of this first sequence of problems is to show that111- = In(2).1-23-45-61a) Given a positive integer n, find an alternative expression for1-r+2n-2- 12n-1.- r° +...(Hint: finite geometric series!)1b) Integrate the result in problem 1 to get an expression for11-21111342n - 12n1c) Show that for all n,111+...+1.an dr.(2n – 1)(2n)dr <r+13-45.6(Hint: you will need that r+1 >1 for r> 0.)1 d) Using Problem 3 and letting n→ 0, show the wanted result.

Answered: Image /qna-images/answer/a31c92cd-798f-461a-80e8-523d894a66e7.jpg

This project is based off of two separate results which show the utility of series and sequences. The first of these is about In(2). The goal of this first sequence of problems is to show that 1 1 1 1-2 - = In(2). 3-4 5-6 Ta) Given a positive integer n, find an alternative expression for 1-r+ 22 – r3 + ... 2n-2 - 2n-1. (Hint: finite geometric series!) 1b) Integrate the result in problem 1 to get an expression for 1 1 - 2 1 1 3 4 2n - 1 2n 1c) Show that for all n, 1 1 1 +..+ 1. (2n – 1)(2n) dr < r+1 2n dr. 3-4 5-6 (Hint: you will need that r+1 >1 for r> 0.) 1 d) Using Problem 3 and letting n+ 0, show the wanted result.

This project is based off of two separate results which show the utility of series and sequences. The first of these is about In(2). The goal of this first sequence of problems is to show that 1 1 1 1-2 - = In(2). 3-4 5-6 Ta) Given a positive integer n, find an alternative expression for 1-r+ 22 – r3 + ... 2n-2 - 2n-1. (Hint: finite geometric series!) 1b) Integrate the result in problem 1 to get an expression for 1 1 - 2 1 1 3 4 2n - 1 2n 1c) Show that for all n, 1 1 1 +..+ 1. (2n – 1)(2n) dr < r+1 2n dr. 3-4 5-6 (Hint: you will need that r+1 >1 for r> 0.) 1 d) Using Problem 3 and letting n+ 0, show the wanted result.

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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