5 pls ## ProblemsIn Problems 1 through 8 find l.u.b. S and g.l.b. S. State whether or not these numbers are in S.1. $ S = \{ x : 0 < x \leq 3 \} $2. $ S = \{ x : x^2 - 3 < 0 \} $3. $ S = \{ x : x^2 - 2x - 3 < 0 \} $4. $ S = \{ y : y = x/(x + 1), x \geq 0 \} $5. $ S = \{ s_n : s_n = \sum_{i=1}^{n} (1/2)^i, n = 1, 2, \ldots \} $6. $ S = \{ s_n : s_n = 1 + \sum_{i=1}^{n} ((-1)^i/i!), n = 1, 2, \ldots \} $7. $ S = \{ x : 0 < x < 5, \cos x = 0 \} $Please solve these problems by finding the least upper bound (l.u.b.) and greatest lower bound (g.l.b.) for each set $ S $. Determine if these bounds are included in the set $ S $.

5) Given set is Noted that Thus we have So we claim that is greast lower bound of is least upper…

Answered: 5. S = {S: S = (1/2'), n = 1, 2,...}

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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5 pls

## Problems

In Problems 1 through 8 find l.u.b. S and g.l.b. S. State whether or not these numbers are in S.

1. $ S = \{ x : 0 < x \leq 3 \} $

2. $ S = \{ x : x^2 - 3 < 0 \} $

3. $ S = \{ x : x^2 - 2x - 3 < 0 \} $

4. $ S = \{ y : y = x/(x + 1), x \geq 0 \} $

5. $ S = \{ s_n : s_n = \sum_{i=1}^{n} (1/2)^i, n = 1, 2, \ldots \} $

6. $ S = \{ s_n : s_n = 1 + \sum_{i=1}^{n} ((-1)^i/i!), n = 1, 2, \ldots \} $

7. $ S = \{ x : 0 < x < 5, \cos x = 0 \} $

Please solve these problems by finding the least upper bound (l.u.b.) and greatest lower bound (g.l.b.) for each set $ S $. Determine if these bounds are included in the set $ S $.

Expert Solution

Step 1: supremum and infimum of a set

5) Given set is $S equals open curly brackets s subscript n semicolon s subscript n equals sum from i equals 1 to n of 1 over 2 to the power of i comma n equals 1 comma 2 comma 3..... close curly brackets$

Noted that $s subscript n equals sum from i equals 1 to n of 1 over 2 to the power of i equals 1 half plus 1 over 2 squared plus..... plus 1 over 2 to the power of n equals 1 half. space fraction numerator 1 minus 1 over 2 to the power of n over denominator 1 minus 1 half end fraction equals 1 minus 1 over 2 to the power of n comma n equals 1 comma 2 comma 3 comma....$

Thus we have $S equals open curly brackets 1 half comma 3 over 4 comma 7 over 8 comma 15 over 16 comma...... close curly brackets$

So we claim that $1 half$ is greast lower bound of $S space a n d space 1 space$ is least upper bound of $S$ .