Ans-11 - Answers, Assignment #11 Exercises 2.2 1. (a)...

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Answers, Assignment #11 Exercises 2.2 1. (a) False. s n = 1 n > 0 for all n ; s n 0. (b) False. { s n } = { 1 , - 1 , 1 , - 1 , ···} ; { t n } = {- 1 , 1 , - 1 , 1 , ···} ; { s n + t n } = { 0 , 0 , 0 ,... }→ 0. (c) False. The example in (b) will work here; { s n t n } = {- 1 , - 1 , - 1 ,... }→- 1. (d) True. Suppose { s n }→ A and { s n + t n }→ B . Then t n =( s n + t n ) - s n B - A . (e) False. Let s n = 1 n ,t n = n . Then { s n t n } = { 1 , 1 , 1 ,... }→ 1, but { t n } diverges. (f) False. Let s n =( - 1) n n = {- 1 , 2 , - 3 , 4 , - 5 ,... } . { s n } is not bounded above, { s n } does not diverge to + . 2. (a) s n →- 2 (b) s n 0 (c) { s n } diverges; for large n, s n ≈± 1 2 (d) s n = 2 3 n 3 2 n = ± 8 9 ² n 0 (e) s n diverges to + (f) s n diverges to + 3. (a) ³ n 2 +1 - n = ³ n 2 +1 - n n 2 +1+ n n 2 +1+ n = 1 n 2 +1+ n 0. (b) ³ n 2 + n - n = ³ n 2 + n - n n 2 + n + n n 2 + n + n = n n 2 + n + n 1 2 . 4. (1) Suppose s n s, t n t . Let ±> 0. Then there exists a positive integer N 1 such that | s n - s | < ± 2 for all n>N 1 . There exists a positive integer N 2 such that | t n - t | < ± 2 for all n>N 2 . Let N = max { N 1 ,N 2 } . Then | ( s n + t n ) - ( s + t ) | = | ( s n - s )+( t n - t ) |≤| s n - s | + | t n - t | < ± 2 + ± 2 = ± for all n>N . (3) Let
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.

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Ans-11 - Answers, Assignment #11 Exercises 2.2 1. (a)...

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