# Convergence Proof: Show {n2 - n + 5} Increasing & {xn} Convergent

In summary, the conversation discusses how to show that the sequence {n^2 - n + 5} is increasing and to use this information to prove that the sequence {x_n} is convergent, where {x_n} = exp[(3n^2 - 3n + 14) / (n^2 - n + 5)]. The conversation also mentions that you may assume that exp x < exp y when x < y, but may not use any properties of the limit of exp x as x → 3. The conversation covers various approaches and the use of the Monotone Sequence Theorem to prove that {x_n} is convergent. It is suggested that an upper bound of e^4 is

## Homework Statement

Show that {n2 - n + 5} is increasing and hence show that {xn} is convergent when

{xn} = exp[(3n2 - 3n +14) / (n2 - n + 5)]

You may assume exp x < exp y when x < y, but may not use any properties of the limit of exp x as x → 3.

## The Attempt at a Solution

I tried to show xn+1 ≥ xn to show {n2 - n + 5} was increasing.

So I got (n+1)2 - (n+1) + 5 ≥ n2 - n + 5

Therefore n2 + n + 5 ≥ n2 - n + 5 and so n ≥ -n

Does this show that {n2 - n + 5} is increasing? Is it enough proof?

Going onto the next bit of the question. I assume because you can't use any properties of the limit of exp x as x → 3 that you must use the Sandwich Theorem? This being that when you have sequences such as:

{zn} ≤ {xn} ≤ {yn}

that if {zn} and {yn} converge, {xn} also must converge.

After this I'm not really too sure what to do, or whether this is actually the right way to approach this question.

Has anyone got any help that they could offer me please?

## Homework Statement

Show that {n2 - n + 5} is increasing and hence show that {xn} is convergent when

{xn} = exp[(3n2 - 3n +14) / (n2 - n + 5)]

You may assume exp x < exp y when x < y, but may not use any properties of the limit of exp x as x → 3.

## The Attempt at a Solution

I tried to show xn+1 ≥ xn to show {n2 - n + 5} was increasing.

So I got (n+1)2 - (n+1) + 5 ≥ n2 - n + 5

Therefore n2 + n + 5 ≥ n2 - n + 5 and so n ≥ -n

Does this show that {n2 - n + 5} is increasing? Is it enough proof?
Kinda. Your steps are backwards. You generally shouldn't start with what you're trying to prove. Reverse the steps, starting with something given, like n>0.

Going onto the next bit of the question. I assume because you can't use any properties of the limit of exp x as x → 3 that you must use the Sandwich Theorem? This being that when you have sequences such as:

{zn} ≤ {xn} ≤ {yn}

that if {zn} and {yn} converge, {xn} also must converge.

After this I'm not really too sure what to do, or whether this is actually the right way to approach this question.

Has anyone got any help that they could offer me please?
You should be able to show the sequence increases monotonically. Do you have any theorems that apply to such a series to show it converges?

1 person

vela said:
Kinda. Your steps are backwards. You generally shouldn't start with what you're trying to prove. Reverse the steps, starting with something given, like n>0.

I'm not really sure how to try to show it's increasing the other way around. I find it difficult because it's obvious looking at it that n2 - n + 5 is increasing but proving this in a formal proof is hard.

I don't think n > 0 necessarily so not sure where to start when reversing the steps like you said but the sequence itself {xn} > 0 as n2 + 5 ≥ n.

vela said:
You should be able to show the sequence increases monotonically. Do you have any theorems that apply to such a series to show it converges?

I know about the Monotone Sequence Theorem which is just that if a sequence is increasing/decreasing and is bounded above/below then it is converging. However, I can't see how you can use it in this case because you aren't allowed to use properties of the limit of exp x as x → 3.

Otherwise you could just divide {xn} = exp[(3n2 - 3n +14) / (n2 - n + 5)] through top and bottom by exp(n2) and it would leave you with a lowest upper bound of e3 and therefore as it's bounded above and is increasing then it must be convergent. I don't think you can do it this way, due to what's stated in the question.

I know about the Monotone Sequence Theorem which is just that if a sequence is increasing/decreasing and is bounded above/below then it is converging. However, I can't see how you can use it in this case because you aren't allowed to use properties of the limit of exp x as x → 3.

Otherwise you could just divide {xn} = exp[(3n2 - 3n +14) / (n2 - n + 5)] through top and bottom by exp(n2) and it would leave you with a lowest upper bound of e3 and therefore as it's bounded above and is increasing then it must be convergent. I don't think you can do it this way, due to what's stated in the question.

You are told to assume that $\exp$ is strictly increasing, so monotone convergence is the correct approach.

What you can't do is say that $\exp$ is continuous, so that
$$\lim_{n \to \infty} \exp(a_n) = \exp\left(\lim_{n \to \infty} a_n\right)$$

Last edited:
1 person
wrong topic post sorry

pasmith said:
You are told to assume that $\exp$ is strictly increasing, so monotone convergence is the correct approach.

What you can't do is say that $\exp$ is continuous, so that
$$\lim_{n \to \infty} \exp(a_n) = \exp\left(\lim_{n \to \infty} a_n\right)$$

I understand that the Monotone Convergence Theorem requires you to show {xn} is increasing and bounded above. However, I'm not sure how you can show it's bounded above without having to show the limit is exp(3).

I understand that the Monotone Convergence Theorem requires you to show {xn} is increasing and bounded above. However, I'm not sure how you can show it's bounded above without having to show the limit is exp(3).

You just need an upper bound; $e^4$ will suffice.

1 person
pasmith said:
You just need an upper bound; $e^4$ will suffice.

Ah right ok so by proving what's inside the brackets for {xn} ≤ 4 will be enough to prove that {xn} is bounded above.

Then use of the monotonic sequence theorem shows {xn} is converging.

I'm not really sure how to try to show it's increasing the other way around. I find it difficult because it's obvious looking at it that n2 - n + 5 is increasing but proving this in a formal proof is hard.

I don't think n > 0 necessarily so not sure where to start when reversing the steps like you said but the sequence itself {xn} > 0 as n2 + 5 ≥ n.
There has to be some condition on n because if you make n negative enough, ##a_n=n^2-n+5## will be decreasing. n is the index of the sequences, so it's (typically) greater than 0. Starting with n>0, you have:

n>0
2n>0
n>-n

and then just list the rest of your steps in reverse order.

The way I would have done it is to calculate what ##a_{n-1}-a_n## is and show the result is positive (with the assumption n>0), from which you can conclude that ##a_{n-1} \gt a_n##.

## 1. What is the definition of convergence?

The concept of convergence refers to the idea that a sequence of numbers or values will approach a certain limit or value as the number of terms in the sequence increases. In other words, the terms in the sequence get closer and closer to a particular value as the sequence progresses.

## 2. How is convergence proven?

Convergence can be proven through various methods, such as the epsilon-delta definition, the Monotone Convergence Theorem, or the Cauchy Criterion. In the case of {n2 - n + 5} increasing and {xn} convergent, it can be proven by showing that the sequence {n2 - n + 5} is bounded above and increasing, and that the limit of {xn} exists and is equal to the limit of {n2 - n + 5}.

## 3. What is the significance of a sequence being increasing?

An increasing sequence is one in which each term is greater than or equal to the previous term. This indicates that the terms are getting larger as the sequence progresses, and can be useful in proving convergence. In the case of {n2 - n + 5} increasing, it means that as the value of n increases, the terms in the sequence also increase, which is a key element in proving convergence.

## 4. What is the importance of showing that a sequence is bounded above?

In order for a sequence to converge, it must be bounded, meaning that the terms in the sequence do not exceed a certain value. In the case of {n2 - n + 5}, showing that it is bounded above means that there is a limit to how large the terms in the sequence can get, which is necessary for proving convergence.

## 5. Why is proving convergence important in mathematics and science?

Proving convergence is important in mathematics and science because it allows us to make predictions and draw conclusions about the behavior of a sequence or function. It is also essential in many fields of research, such as physics, engineering, and economics, where understanding and predicting the behavior of systems or processes is crucial. Additionally, convergence proofs can help us identify patterns and relationships between different sequences or functions, leading to new discoveries and advancements in various fields.

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