Question

4.2.7. Example. If (xn) is a sequence in (0, ∞) and xn → a, then √xn...

4.2.7. Example. If (xn) is a sequence in (0, ∞) and xn → a, then √xn → √a.
?
Proof. Problem. Hint. There are two possibilities: treat the cases a = 0 and a > 0 sepa- √√
rately. For the first use problem 4.1.7(a). For the second use 4.2.1(b) and 4.1.11; write xn − a as |xn − a|/(√xn + √a). Then find an inequality that allows you to use the sandwich theo- rem(proposition 4.2.5).
Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

Here is the required proof.I hope the solution will help you.Expecting a thumbs up if you are satisfied with the work,it will help me a lot.Thank you.

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
) Let α be a fixed positive real number, α > 0. For a sequence {xn},...
) Let α be a fixed positive real number, α > 0. For a sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that {xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all n). (b) Prove that {xn} is bounded from below. (Hint: use proof by induction to show xn > √ α for all...
1. Give a direct proof that the product of two odd integers is odd. 2. Give...
1. Give a direct proof that the product of two odd integers is odd. 2. Give an indirect proof that if 2n 3 + 3n + 4 is odd, then n is odd. 3. Give a proof by contradiction that if 2n 3 + 3n + 4 is odd, then n is odd. Hint: Your proofs for problems 2 and 3 should be different even though your proving the same theorem. 4. Give a counter example to the proposition: Every...
Consider n independent variables, {X1, X2, . . . , Xn} uniformly distributed over the unit...
Consider n independent variables, {X1, X2, . . . , Xn} uniformly distributed over the unit interval, (0, 1). Introduce two new random variables, M = max (X1, X2, . . . , Xn) and N = min (X1, X2, . . . , Xn). (A) Find the joint distribution of a pair (M, N). (B) Derive the CDF and density for M. (C) Derive the CDF and density for N. (D) Find moments of first and second order for...
Fibonacci Sequence (Javascript in HTML)   function fib(first, second, countdown){     console.log("I was called with " + first...
Fibonacci Sequence (Javascript in HTML)   function fib(first, second, countdown){     console.log("I was called with " + first + ", " + second + ", " + countdown);         if(countdown>0){                                 fib(first,first+second, countdown-1);         }     }     fib(1,2,3); Now you get to figure out how to use the fib function to print the number sequence without using a loop in the function (hint, the function needs to call itself). How is this exactly done? It needs to be in written so I can input it into...
Suppose that Newton’s method is applied to find the solution p = 0 of the equation...
Suppose that Newton’s method is applied to find the solution p = 0 of the equation e^x −1−x− (1/2)x^2 = 0. It is known that, starting with any p0 > 0, the sequence {pn} produced by the Newton’s method is monotonically decreasing (i.e., p0 >p1 >p2 >···)and converges to 0. Prove that {pn} converges to 0 linearly with rate 2/3. (hint: You need to have the patience to use L’Hospital rule repeatedly. ) Please do the proof.
1. Let X1, . . . , Xn be i.i.d. continuous RVs with density pθ(x) =...
1. Let X1, . . . , Xn be i.i.d. continuous RVs with density pθ(x) = e−(x−θ), x ≥ θ for some unknown θ > 0. Be sure to notice that x ≥ θ. (This is an example of a shifted Exponential distribution.) (a) Set up the integral you would solve for find the population mean (in terms of θ); be sure to specify d[blank]. (You should set up the integral by hand, but you can use software to evaluate...
Consider the function g (x) = 12x + 4 - cos x. Given g (x) =...
Consider the function g (x) = 12x + 4 - cos x. Given g (x) = 0 has a unique solution x = b in the interval (−1/2, 0), and you can use this without justification. (a) Show that Newton's method of starting point x0 = 0 gives a number sequence with b <··· <xn+1 <xn <··· <x1 <x0 = 0 (The word "curvature" should be included in the argument!) (b) Calculate x1 and x2. Use theorem 2 in section...
Consider the function g (x) = 12x + 4 - cos x. Given g (x) =...
Consider the function g (x) = 12x + 4 - cos x. Given g (x) = 0 has a unique solution x = b in the interval (−1/2, 0), and you can use this without justification. (a) Show that Newton's method of starting point x0 = 0 gives a number sequence with b <··· <xn+1 <xn <··· <x1 <x0 = 0 (The word "curvature" should be included in the argument!) (b) Calculate x1 and x2. Use theorem 2 in section...
Consider the function g (x) = 12x + 4 - cos x. Given g (x) =...
Consider the function g (x) = 12x + 4 - cos x. Given g (x) = 0 has a unique solution x = b in the interval (−1/2, 0), and you can use this without justification. (a) Show that Newton's method of starting point x0 = 0 gives a number sequence with b <··· <xn+1 <xn <··· <x1 <x0 = 0 (The word "curvature" should be included in the argument!) (b) Calculate x1 and x2. Use theorem 2 in section...
1. Write 3n + 1 Sequence String Generator function (1 point) In lab5.py complete the function...
1. Write 3n + 1 Sequence String Generator function (1 point) In lab5.py complete the function sequence that takes one parameter: n, which is the initial value of a sequence of integers. The 3n + 1 sequence starts with an integer, n in this case, wherein each successive integer of the sequence is calculated based on these rules: 1. If the current value of n is odd, then the next number in the sequence is three times the current number...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT