Convergence of Sequence Proof: Is This Correct?

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.
  • #1
Mr Davis 97
1,462
44

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?
 
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  • #3
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?
 
  • #4
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##.
 
  • #5
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.
 
  • #6
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?
 
  • #8
@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.
 

Related to Convergence of Sequence Proof: Is This Correct?

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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