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Using only the definition of limit (so, without using Arithmetic of Limits), show that limn→∞ 2/n + 3/(n+1) = 0

Using only the definition of limit (so, without using Arithmetic of Limits), show that limn→∞ 2/n + 3/(n+1) = 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 81E
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B. Using only the definition of limit (so, without using Arithmetic of Limits), show that
limn→∞ 2/n + 3/(n+1) = 0

Example: Show directly from definition that lim Y/₁² = 0₁ 510 Sulution 1: Proceed exactly similarly to Solution 2: Notice that for nz1, Thus, With more and 1²³/n²-0] = 1¹/₁² <½/n, 50 we take N= Y/₂ After some algebra, and complicated sequences, this technique of "replacing" (bounding) w/ something simpler be quite useful. Examples Show directly from definition We want to find conditions Solution: 2n²+ 34 -21<ε. n²-2 sume algebra, (*) can So In ²-21 = 23/1²= n Multiplying together, We consider the top + buttom separately! 13n+41 ≤ 13n+4n1=17n1 In²-21 = | 11/²2/1 Examples show that the sequence Solution: Suppose that Then Yn² ≤ ½n, lim |3n++ | ≤ 7n. 3/2 = 210 22 n²2 So N=21/₂ 336 Cun Remarks This last example will become that we will prove (We showed earlier that 316 VETO, IN sit. that so that becomes | 3n++ | <E. Sn=S₁ when nz3 (as then when n23. lim 316 (as in the Y/n example). when nz3. will suffice, by "sulving" as in previous examples. lim 2n²+ 3n n²-2 316 when n21 Sn=(-+)" does not converge. Sn # 1,) 2 = 21² ) 37 =2. much easier with theorems bit later (*) [n>N] => [1(-1)" - 5) <ε].
Transcribed Image Text:Example: Show directly from definition that lim Y/₁² = 0₁ 510 Sulution 1: Proceed exactly similarly to Solution 2: Notice that for nz1, Thus, With more and 1²³/n²-0] = 1¹/₁² <½/n, 50 we take N= Y/₂ After some algebra, and complicated sequences, this technique of "replacing" (bounding) w/ something simpler be quite useful. Examples Show directly from definition We want to find conditions Solution: 2n²+ 34 -21<ε. n²-2 sume algebra, (*) can So In ²-21 = 23/1²= n Multiplying together, We consider the top + buttom separately! 13n+41 ≤ 13n+4n1=17n1 In²-21 = | 11/²2/1 Examples show that the sequence Solution: Suppose that Then Yn² ≤ ½n, lim |3n++ | ≤ 7n. 3/2 = 210 22 n²2 So N=21/₂ 336 Cun Remarks This last example will become that we will prove (We showed earlier that 316 VETO, IN sit. that so that becomes | 3n++ | <E. Sn=S₁ when nz3 (as then when n23. lim 316 (as in the Y/n example). when nz3. will suffice, by "sulving" as in previous examples. lim 2n²+ 3n n²-2 316 when n21 Sn=(-+)" does not converge. Sn # 1,) 2 = 21² ) 37 =2. much easier with theorems bit later (*) [n>N] => [1(-1)" - 5) <ε].
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