# Convergence of Sequence Proof: Is This Correct?

• Mr Davis 97
In summary, the conversation discusses proving the convergence of a sequence using the definition of convergence. The solution presented involves choosing an appropriate value for N, with the concern raised that N should be an integer according to the definition. It is ultimately concluded that it doesn't matter if N is an integer as long as n > N is an integer.
Mr Davis 97

## Homework Statement

Prove rigorously that ##\displaystyle \lim \frac{n}{n^2 + 1} = 0##.

## Homework Equations

A sequence ##(s_n)## converges to ##s## if ##\forall \epsilon > 0 \exists N \in \mathbb{N} \forall n \in \mathbb{N} (n> N \implies |s_n - s| < \epsilon)##

## The Attempt at a Solution

Let ##\epsilon > 0##. Let ##\displaystyle N = \frac{1}{\epsilon}##. Let ##n \in \mathbb{N}##.
Then, if ##n > N##, we have that ##\displaystyle n > \frac{1}{\epsilon}## and so ##\displaystyle \frac{1}{n} < \epsilon##. Therefore, ##\displaystyle |\frac{n}{n^2+1} - 0| = \frac{n}{n^2+1} < \frac{n}{n^2} = \frac{1}{n} < \epsilon##. This proves that ##\displaystyle \lim \frac{n}{n^2 + 1} = 0##.

Is this a correct convergence proof?

Looks OK to me.

LCKurtz said:
Looks OK to me.
I actually have a concern. I let ##\displaystyle N = \frac{1}{\epsilon}##, but by the definition of sequence convergence, shouldn't ##N## be an integer? Is this a problem? How could I fix it?

Mr Davis 97 said:
I actually have a concern. I let ##\displaystyle N = \frac{1}{\epsilon}##, but by the definition of sequence convergence, shouldn't ##N## be an integer? Is this a problem? How could I fix it?
Take N = ##\lceil \frac 1 \epsilon \rceil##, IOW, the smallest integer greater than or equal to ##\frac 1 \epsilon##.

Mr Davis 97 said:
I actually have a concern. I let ##\displaystyle N = \frac{1}{\epsilon}##, but by the definition of sequence convergence, shouldn't ##N## be an integer? Is this a problem? How could I fix it?

Nothing requires N to be an integer. It is the n > N that is an integer.

LCKurtz said:
Nothing requires N to be an integer. It is the n > N that is an integer.
Well my definition claims that ##N## is a natural number. Would Mark44's approach suffice?

@Mr Davis 97, you're overthinking this. If ##\frac 1 \epsilon## isn't an integer, just take N to be the next-larger integer, which is basically what I said before. After all, N is just a lower bound on the indexes of the sequence elements that are "small enough." Elements further in the sequence will be even smaller; i.e., closer to zero.

## 1. What is convergence of a sequence?

Convergence of a sequence refers to the behavior of a sequence of numbers as its terms approach a specific value or limit. It describes the tendency of the terms in a sequence to get closer and closer to a single value as the sequence progresses.

## 2. How is convergence of a sequence determined?

The convergence of a sequence is determined by evaluating the limit of the sequence. If the limit exists and is a finite value, then the sequence is said to converge. If the limit does not exist or is infinite, then the sequence is said to diverge.

## 3. What is the difference between convergence and divergence?

Convergence and divergence are opposite concepts when talking about the behavior of a sequence. Convergence refers to the tendency of a sequence to approach a single value, while divergence describes the behavior of a sequence that does not approach a single value or tends to infinity.

## 4. How is convergence of a sequence related to the terms in the sequence?

The convergence of a sequence is directly related to the behavior of its terms. If the terms in a sequence get closer and closer to a single value, then the sequence is said to converge. If the terms do not approach a single value, then the sequence is said to diverge.

## 5. What are some real-life applications of convergence of a sequence?

The concept of convergence of a sequence is widely used in various scientific and mathematical fields. Some real-life applications include predicting the behavior of financial markets, analyzing the behavior of chemical reactions, and studying the properties of infinite series in physics and engineering.

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