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
Rina8888 [55]
3 years ago
6

Consider the sequence {an}={3n+13n−3n3n+1}. Graph this sequence and use your graph to help you answer the following questions.

Mathematics
1 answer:
Fantom [35]3 years ago
5 0

Part 1: You can simplify a_n to

\dfrac{3n+1}{3n}-\dfrac{3n}{3n+1} = \dfrac1{3n}+\dfrac1{3n+1}

Presumably, the sequence starts at <em>n</em> = 1. It's easy to see that the sequence is strictly decreasing, since larger values of <em>n</em> make either fraction smaller.

(a) So, the sequence is bounded above by its first value,

|a_n| \le a_1 = \dfrac13+\dfrac14 = \boxed{\dfrac7{12}}

(b) And because both fractions in a_n converge to 0, while remaining positive for any natural number <em>n</em>, the sequence is bounded below by 0,

|a_n| \ge \boxed{0}

(c) Finally, a_n is bounded above and below, so it is a bounded sequence.

Part 2: Yes, a_n is monotonic and strictly decreasing.

Part 3:

(a) I assume the choices are between convergent and divergent. Any monotonic and bounded sequence is convergent.

(b) Since a_n is decreasing and bounded below by 0, its limit as <em>n</em> goes to infinity is 0.

Part 4:

(a) We have

\displaystyle \lim_{n\to\infty} \frac{10n^2+1}{n^2+n} = \lim_{n\to\infty}10+\frac1{n^2}}{1+\frac1n} = 10

and the (-1)ⁿ makes this limit alternate between -10 and 10. So the sequence is bounded but clearly not monotonic, and hence divergent.

(b) Taking the limit gives

\displaystyle\lim_{n\to\infty}\frac{10n^3+1}{n^2+n} = \lim_{n\to\infty}\frac{10+\frac1{n^3}}{\frac1n+\frac1{n^2}} = \infty

so the sequence is unbounded and divergent. It should also be easy to see or establish that the sequence is strictly increasing and thus monotonic.

For the next three, I'm guessing the options here are something to the effect of "does", "may", or "does not".

(c) may : the sequence in (a) demonstrates that a bounded sequence need not converge

(d) does not : a monotonic sequence has to be bounded in order to converge, otherwise it grows to ± infinity.

(e) does : this is true and is known as the monotone convergence theorem.

You might be interested in
If AP 10 - 12. what is the perimeter of quadrilateral TAEO (a) 56 (b)64 (c)72 (d)76​
BlackZzzverrR [31]

Husofjwiebd9ks ueijddlonf durnkofbjus yuebejuiiiiiiiiijjd dijdjdjdnsjduwjwuwhw

4 0
3 years ago
Solve this equation for A: A ÷ 2 = 4
Natali5045456 [20]
The answer to the equation is 8
5 0
3 years ago
Read 2 more answers
Tammy and Wyatt are sales associates at the same used car dealership. Their supervisor is planning to promote the employee with
lisov135 [29]
I think the answer is D. The other answers are unreasonable. Wyatt would not get the promotion and just because Tammy's lowest value is bigger than Wyatt's doesn't mean that Tammy gets the promotion. The only reasonable answer is D.

7 0
3 years ago
What is the image point (0,2) after a rotation of 90 degree counterclockwise about the origin
Sidana [21]

Answer:

I like to do mehndi is known as

.

5 0
2 years ago
Read 2 more answers
A square pyramid has side lengths each measuring 8 centimeters. The height of the pyramid is 3 centimeters,
tino4ka555 [31]
You answer would be 80 cm^2
7 0
3 years ago
Other questions:
  • Express 42 and 150 as a product of its prime numbers.
    15·1 answer
  • 7/2(3*2)/(2/(8-6))= ??
    10·1 answer
  • 8 + x/2 = 12 need quick ☁
    10·1 answer
  • Explain how to use a number line to show that opposite quantities combine to make 0
    6·1 answer
  • Add or subtract. –16 + (-14) + (-35)
    15·2 answers
  • 6 pizzas shared equally. Each child got 1/9 of a pizza. How many children were in the group?
    5·2 answers
  • Advance pile up trigonometry
    9·1 answer
  • 5(y +2/5) = -13
    6·1 answer
  • Find an equation for the line below <br> (Pls provide explanation)
    10·1 answer
  • a) The difference between the circumference and the diameter of a circle is 90cmFind the radius of the circle​
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!