Proof by induction: Base case: trynna see if the sequence is increasing. n=1 Xn € Xn+1 by definition X₁ = × 1+1 VEN x₁ = X2 √ ≤ 2 ✓ Induction Steps: Let n = k Э KEIN we know to show that and xx = Xxxl we are trying Xx+1 £ xk+2 Therefore, +2 (πx) = (xx+1)+3 √2+x=12+ X1 Xx+1 = Xk+2 Thus we can conclude that the sequence {X} Sequence. by the monotone definition is an increasing Now problem we know we need to show if it 15 bounded in the X ≤ 2 and given √2 = Proof by Induction: Base case: we know √2 ≤ X ≤ 2, UnEN √2 ± X₁ = 2 √2 ± √2 ± 2 ✓ Induction Steps: Let n = k B KEN ≤ and we want to we know √2 = Xx 2 Show that So. Xx+ 1 ≤ 2. +2 (√2)+2(x)(2) +2 √2+82 €12+ x =√4 2≤ Укн ≤2 Therefore sequence {Xx} In conclusion n In base on this we know that the is bounded. Show that the sequence {K^} is Therefore has the definition we d bounded In conclusion increasing and ' we show that the sequence 9x] 15 and bounded. Therefore by the definition of monotone convergence {v} is converges. thereom this sequence Now we need to find the limit: * Let X" be the limit lum (date) = √2+ km (x) 1im (xn+1) 7*2 +#2 - χ 2 2+ 2 + x* x-2=0 (x*+1) (x* -2) X=-1 = 0 x = 2 ** The limit is 2 2. Let the sequence (xn) be recursively defined by x₁ = √√2 and Xn+1 = √2+xn, n≥1. Show that (xn) converges and evaluate its limit.
Proof by induction: Base case: trynna see if the sequence is increasing. n=1 Xn € Xn+1 by definition X₁ = × 1+1 VEN x₁ = X2 √ ≤ 2 ✓ Induction Steps: Let n = k Э KEIN we know to show that and xx = Xxxl we are trying Xx+1 £ xk+2 Therefore, +2 (πx) = (xx+1)+3 √2+x=12+ X1 Xx+1 = Xk+2 Thus we can conclude that the sequence {X} Sequence. by the monotone definition is an increasing Now problem we know we need to show if it 15 bounded in the X ≤ 2 and given √2 = Proof by Induction: Base case: we know √2 ≤ X ≤ 2, UnEN √2 ± X₁ = 2 √2 ± √2 ± 2 ✓ Induction Steps: Let n = k B KEN ≤ and we want to we know √2 = Xx 2 Show that So. Xx+ 1 ≤ 2. +2 (√2)+2(x)(2) +2 √2+82 €12+ x =√4 2≤ Укн ≤2 Therefore sequence {Xx} In conclusion n In base on this we know that the is bounded. Show that the sequence {K^} is Therefore has the definition we d bounded In conclusion increasing and ' we show that the sequence 9x] 15 and bounded. Therefore by the definition of monotone convergence {v} is converges. thereom this sequence Now we need to find the limit: * Let X" be the limit lum (date) = √2+ km (x) 1im (xn+1) 7*2 +#2 - χ 2 2+ 2 + x* x-2=0 (x*+1) (x* -2) X=-1 = 0 x = 2 ** The limit is 2 2. Let the sequence (xn) be recursively defined by x₁ = √√2 and Xn+1 = √2+xn, n≥1. Show that (xn) converges and evaluate its limit.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 8RE
Question
Im trying to solve this problem based on what i known but i feel like my proof to check to see if the sequence is bounded is incorrect so can someone please take a look see if i did it right and also the whole proof for the problem please? Thank you very much.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning