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
Decimal equivalent of 143%
daser333 [38]

Answer:

1.43 as a decimal

Step-by-step explanation:

%143 = 1.43

7 0
2 years ago
Hey guys i really need help pls dont waste the points i actually want help to understand how to solve equations like:
tamaranim1 [39]

Answer:

your question is unclear

Step-by-step explanation:

x+2y=4

-x -x

2y=4-x

divide by 2 to get y by it's self

y=2-(1/2x)

x+2y=4

-2y -2y

x=4-2y

i don't know if u want me to help u with the second one but i did it the first one

3 0
3 years ago
The Wongs are putting a flagstone deck around their pool. The pool is rectangular. Its dimensions are 8 m by 4 m. The deck will
VLD [36.1K]

Answer: 64 i believe

Hope this helps :)

3 0
3 years ago
Hurry if correct I will give brainly ​
kow [346]

Answer:

D

Step-by-step explanation:

A=20*20

6 0
3 years ago
Given that 3 feet equal 1 yard, how many yards equals 1 feet ?
never [62]

Answer:

0.333333

Step-by-step explanation:

Hope this helps and have a great day!

6 0
3 years ago
Other questions:
  • Joe biked the following distances this past week to train for a triathlon. Joe biked 37 miles, 88 miles, 44 miles, 54 miles, 62
    13·1 answer
  • What number belongs in the blank space below
    8·1 answer
  • Ask Your Teacher Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified
    6·1 answer
  • Does anyone know how many ways you can make 16 and 19 on a grid
    15·1 answer
  • How to I find this?<br><br> What percent of 60 is 42?
    13·2 answers
  • Solve for y thank you
    14·1 answer
  • You have orders to give a patient 20 mg of a certain medication. The medication is stored 4 mg per 5 mL dose. How many millilite
    8·2 answers
  • A submarine started 20 feet below sea level. It is now 10 feet below sea level. Which integer represents its change in depth?
    5·2 answers
  • Determine whether each statement is true or false.
    7·1 answer
  • Marco bounces a basketball 45 times in 60 seconds.at the rate approximately how many times will Marco bounce the ball 150 secon
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!