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
Ellie buys a mobile phone in spain for €352.50. Her brother jack buys an identical phone from japan for ¥39856. Using exchange r
Nonamiya [84]

Answer:

The answer to your question is: Japan

Step-by-step explanation:

Mobile cost in Spain = € 352.5

Mobile cost in Japan = ¥39856

Exchange rate = £1 = €1.41

                           £1 = ¥188

Cost of mobile in pounds

                       £ 1 ------------------ € 1.41

                        x  -----------------   €352.5

                       x = (352.5 x 1) / 1.41

                       x= £250

                      £1   ------------------  ¥188

                      x    ----------------- -¥ 39856

                     x = (39856 x 1) / 188

                     x = £212

Then, it was cheaper in Japan

8 0
3 years ago
Read 2 more answers
A bag contains 5 grey marbles, 7 striped marbles, and 3 black marbles. If one marble is randomly chose, what is the probability
Anvisha [2.4K]

Answer:

C

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
Determine the intercepts of the line. x-intercept:( , ) y-intercept: ( , )​
sasho [114]

Answer:

y intercept is -10

x intercept is 3

the number at which the line intersects with the line is what it will be. Remember that the x intercept is horizontal and the y intercept is vertical

7 0
3 years ago
Write the standard form of the equation of the line through the pair of points (6,3) and (-7,7)
timofeeve [1]

Step-by-step explanation:

steps are in the picture above.

<h3><u>N</u><u>o</u><u>t</u><u>e</u><u>:</u><u>i</u><u>f</u><u> </u><u>y</u><u>o</u><u>u</u><u> </u><u>n</u><u>e</u><u>e</u><u>d</u><u> </u><u>t</u><u>o</u><u> </u><u>a</u><u>s</u><u>k</u><u> </u><u>a</u><u>n</u><u>y</u><u> </u><u>question</u><u> </u><u>please</u><u> </u><u>let</u><u> </u><u>me</u><u> </u><u>know</u><u>.</u></h3>

3 0
3 years ago
What is the volume of 5cm 2cm 8cm.
Soloha48 [4]

Answer:

80 cm

Step-by-step explanation:

(5 x 8) x 2

40 x 2

80 cm

5 0
3 years ago
Read 2 more answers
Other questions:
  • Someone help please
    10·1 answer
  • If gas costs $3.95 per gallon at a local service station, how many cubic feet of gas can a customer buy with $50
    5·1 answer
  • PLEASE HELP!!!!! And explain too! Thanks!!!
    6·1 answer
  • 5/12 as a whole number
    9·2 answers
  • Which of the following equations is equivalent to 2(3x - 1) = 3 - (- 2 - x)?
    14·2 answers
  • If there were 125 students in 5 buses what is the ratio of students to buses
    13·1 answer
  • ASAP PLEASE
    8·2 answers
  • What is an equation of the line that passes through the point (-2,-3)(−2,−3) and is parallel to the line 5x+2y=14
    8·1 answer
  • Joe brought 17 packs of cola to a party, and each pack had 8 cans. Teresa brought 6 cans of juice. How many cans did they bring
    10·2 answers
  • O. The number of for attending football match this week was 12% more more than last week. if 728 people attended this week's mat
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!