1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
myrzilka [38]
3 years ago
11

Find the number to which the sequence {(3n+1)/(2n-1)} converges and prove that your answer is correct using the epsilon-N defini

tion of convergence.
Mathematics
1 answer:
Nat2105 [25]3 years ago
6 0
By inspection, it's clear that the sequence must converge to \dfrac32 because

\dfrac{3n+1}{2n-1}=\dfrac{3+\frac1n}{2-\frac1n}\approx\dfrac32

when n is arbitrarily large.

Now, for the limit as n\to\infty to be equal to \dfrac32 is to say that for any \varepsilon>0, there exists some N such that whenever n>N, it follows that

\left|\dfrac{3n+1}{2n-1}-\dfrac32\right|

From this inequality, we get

\left|\dfrac{3n+1}{2n-1}-\dfrac32\right|=\left|\dfrac{(6n+2)-(6n-3)}{2(2n-1)}\right|=\dfrac52\dfrac1{|2n-1|}
\implies|2n-1|>\dfrac5{2\varepsilon}
\implies2n-1\dfrac5{2\varepsilon}
\implies n\dfrac12+\dfrac5{4\varepsilon}

As we're considering n\to\infty, we can omit the first inequality.

We can then see that choosing N=\left\lceil\dfrac12+\dfrac5{4\varepsilon}\right\rceil will guarantee the condition for the limit to exist. We take the ceiling (least integer larger than the given bound) just so that N\in\mathbb N.
You might be interested in
Help please, ill give 5 stars, and brainly ofc (if correct)
lesya [120]

Answer:

Step-by-step explanation:

part a = 1 x 4

part b = b

5 0
2 years ago
Read 2 more answers
Four square pieces of corkboard are lined up to make a rectangle. The area of the entire corkboard is
Pavel [41]

9514 1404 393

Answer:

  a) attached

  b) 100 square inches, 1/4 of the total

  c) square root of area

  d) multiply by 4

Step-by-step explanation:

a) A diagram is attached

__

b) If the total area consists of 4 squares of the same size, then the area of one square will be 1/4 of the total area. 1/4 of 400 square inches is 100 square inches, the area of one square.

__

c) The area of a square is the square of the side length. You find the side length by finding the square root of the area. √(100 in²) = 10 in, the side length of each square.

__

d) The rectangle is 4 squares laid side by side, so its length is 4 times the side length of one square: 4×10 in = 40 in.

4 0
3 years ago
The quotient of a number and 4 is greater than or equal to 39
bonufazy [111]

x/4 ≥ 9    is the expression you want

4 0
3 years ago
Megan is sitting in her car in a parking lot. The car next to her pulls forward. If Megan is looking only at the car next to her
Zina [86]

Answer:Backward

Step-by-step explanation:

If the car next to Megan is being Pulled Forward and Megan is seeing only the pulled car then her car appears to be moving backward.

This can be said by using the concept of relative velocity which states that the speed of the body with respect to the other is considered to be at rest.

Here Megan's car appears to move backward with respect to the car next to her due to the existence of relative velocity.

6 0
2 years ago
Y=4x+8forx <br> Thank you if you can answer
yKpoI14uk [10]

Answer:

y=4x-8. y=4x−8 y = 4 x -

Step-by-step explanation:

I think is it

7 0
2 years ago
Read 2 more answers
Other questions:
  • Does a negative minus a negative equal positive
    10·2 answers
  • What is the numerator of the simplified difference? 2x+3/x+4 - x-7/x+4​
    6·1 answer
  • Let f(x) = x − 2 and g(x) = x2 − 7x − 9. Find f(g(−1)). (1 point)
    10·2 answers
  • 2.Mrs. Fernandez and Mrs.Bosiljevac invaded the biology lab! They find some bacterial cultures and decide to experiment. Mrs. Fe
    6·1 answer
  • Gina brought 6 containers of yogurt for $4.how many of yogurt could Gina buy for $12
    11·2 answers
  • Can somewon help me i eally need help
    7·2 answers
  • Compare and arrange each
    10·1 answer
  • Which of the following is a factor from problem #1?<br><br> 1) X^2 -3x -10
    7·1 answer
  • Distributive property 21x4
    11·1 answer
  • Please help me with this!!!
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!