1549901369-Elements_of_Real_Analysis__Denlinger_

642 Appendix C 11 Answers & Hints for Selected Exercises

Xn = ~ -; 0. (Apply Exercise 2.2.16.) When lei < 1,

nP

I

Xn+l \ = I ( en+\. nP I= lei (-n-)P = lcl (1 + ~)P-; lei < 1, so by

Xn n + 1 P en n + 1 n

nP

Thm. 2.3. 10 , Xn-; 0. (b) Do as in (a) but with Xn = -en.

8. 0 :S: a :S: b ~ 0 :S: bn :S: an + bn :S: 2bn ~ b :S: \I an + bn :S: f/2b. Apply

Example 2.3.9 and squeeze.

11. If e = 0, { ~~} is a constant sequence. If e 'f. 0 let Xn = ~~. Then

I

--Xn+l I = I ( en+l )' · -n! I = --lei -t 0. Apply Thm. 2.3.10.

Xn n + 1. en n + 1

n

13. Let x > 1, and 't:/n EN, Xn =-.Then
    xn

I

--Xn+l I = --n +^1 · xn - = n --+ 1 · -^1 -t -^1 < 1. Apply Thm. 2.3.10.

Xn xn+l n n X X

15. O; O; - 1; O; O; ~; O; O; O; 0


16. If e = 0, {nPen} is a constant sequence. If 0 < lcl < 1, let Xn = nPen.

Then I x~:

1

/ = I (n +n~~nen+l I = lei ( n:

1

) P = lei ( 1 + ~) P-; lei < 1.

Apply Thm. 2.3.10.

20. (a) en - 1 < lcnJ :S: en, so e - * < l';J ::::; e.

(b) Since c~ -; 0, :lno EN 3 n ~no~ 0 < c~ < 1 ~ lc~J = 0.

EXERCISE SET 2.4

1. (a) 10 , 000 , M^2 (b) 101 , M + 1 (c) 10 , 201 , (M + 1)^2
   (d) 18 , v3M + 3.
   Hint for the proof for ( c): to achieve^1 ;?n < -M, or n;,{ > M, note that

n-1 fa > fa+l n-1 = (fa+l)(fa-l) fa+l = y,. 'n - (^1) ' so take y 'n '" - 1 > M.
3n+sn 1 7n 7n (7)n

3. (a) Let Xn =. then 0 < - = < - < - -; 0 by
   7n Xn 3n + 8n 8n 8
   Thm. 2.3.7. Apply the squeeze principle and Thm. 2.4.4.
   In (b) we show an alternative approach, using the comparison test (2.4.7).

n^3 + sin(3n) n^3 - 1

(b)

1

~ --= n^2 +n+1 > n-; oo. Apply Thm. 2.4.7.

n- n-l