Exact upper and lower limit of the sequence

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Discussion Overview

The discussion revolves around finding the exact upper and lower limits of the sequence defined by \( a_n = \frac{2n+3}{n} \). Participants explore the mathematical properties of the sequence, including its behavior as \( n \) increases, and seek to establish bounds and proofs for these limits.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant rewrites the sequence as \( a_n = 2 + \frac{3}{n} \) and prompts others to consider the behavior of the terms as \( n \) grows.
  • Another participant suggests that the upper limit is 5 and the lower limit is 2, but seeks a mathematical proof for these claims.
  • There is a discussion about how the bounds were determined, with one participant noting that \( a_1 = 5 \) and that \( \frac{3}{n} \) decreases as \( n \) increases, indicating that \( a_n \) is monotonically decreasing.
  • A participant incorrectly states that the limit as \( n \) approaches infinity is 0, which is corrected by another participant who clarifies that the limit is actually 2.
  • One participant expresses confidence in having established the lower limit through limits and acknowledges the upper limit as 5, while seeking guidance on how to present this correctly with proof.
  • Another participant suggests discussing the monotonic nature of the sequence and considering the difference \( \frac{3}{n+1} - \frac{3}{n} \) to demonstrate that it is negative for all \( n \in \mathbb{N} \).

Areas of Agreement / Disagreement

Participants generally agree on the lower limit being 2 and the upper limit being 5, but there is no consensus on the best way to prove these limits mathematically.

Contextual Notes

There are unresolved aspects regarding the formal proof of the upper limit and the clarity of the reasoning behind the established bounds. The discussion reflects varying levels of understanding and assumptions about the sequence's behavior.

theakdad
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I have one more problem,for the given sequence i have to find the exact upper and lower limit,and to argument them. i have been missing on this lesson,so please help me,i don't know how to do it.

So the sequence is:

an = $$\frac{2n+3}{n}$$
 
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If I am interpreting correctly what you meant by upper and lower limit, I would rewrite the $n$th term as follows:

$$a_n=2+\frac{3}{n}$$

What can you say about the terms as $n$ grows?
 
MarkFL said:
If I am interpreting correctly what you meant by upper and lower limit, I would rewrite the $n$th term as follows:

$$a_n=2+\frac{3}{n}$$

What can you say about the terms as $n$ grows?

I would say that upper limit is 5 and lower limit is 2.
But how to prove it? Or show it in math way?
 
How did you determine the bounds?
 
MarkFL said:
How did you determine the bounds?

I have assumed for a1
 
wishmaster said:
I have assumed for a1

You know:

$$a_1=2+\frac{3}{1}=5$$

and you know that $$\frac{3}{n}$$ gets smaller as $n$ increases, so then you know $a_n$ is monotonically decreasing. How can you determine the lower bound?
 
MarkFL said:
You know:

$$a_1=2+\frac{3}{1}=5$$

and you know that $$\frac{3}{n}$$ gets smaller as $n$ increases, so then you know $a_n$ is monotonically decreasing. How can you determine the lower bound?

$$\lim _{n \to \infty}2+ \frac{3}{n} = 2 + 0=0$$ ?
 
wishmaster said:
$$\lim _{n \to \infty}2+ \frac{3}{n} = 2 + 0=0$$ ?

Correct, except that $2+0=2$.
 
MarkFL said:
Correct, except that $2+0=2$.
Im sorry,thats what i thought..my mistake.

So with limit i have proved exact lower limit,i know that exact upper limit is 5,but how to write it correctly? With proof...
 
  • #10
wishmaster said:
Im sorry,thats what i thought..my mistake.

So with limit i have proved exact lower limit,i know that exact upper limit is 5,but how to write it correctly? With proof...

What more do you need? You know the sequence bounds and that it is monotonic. All you need to do is discuss the fact that $\dfrac{3}{n}$ decreases as $n$ increases. Or you could consider the difference:

$$\frac{3}{n+1}-\frac{3}{n}$$

And show that it is negative for all $n\in\mathbb{N}$.
 

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