Need guidance on my proof of limit.

In summary, the sequence is increasing and bounded, and therefore the lim_{n\rightarrow +\infty} { (1 + 1/(2n))^n } exists.
  • #1
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Homework Statement


Show that [tex] lim_{n\rightarrow +\infty} { (1 + 1/(2n))^n } = \sqrt{e}[/tex].

Homework Equations


I am allowed to assume * [tex] lim_{n\rightarrow +\infty} { (1 + 1/n )^n} = e[/tex].
I am not allowed to use the theorem that asserts [tex]lim_{n\rightarrow n_0} {\sqrt{S_n}} = (lim_{n\rightarrow n_0} {S_n})^{1/2} [/tex].

The Attempt at a Solution



I want to show that the sequence is increasing and bounded, and therefore the [tex]lim_{n\rightarrow +\infty} { (1 + 1/(2n))^n }[/tex] exists. Let's suppose I have show the sequence is increasing by comparing [tex] S_k[/tex] and [tex] S_{k+1} [/tex] and showing [tex] \forall k, S_k < S_{k+1}[/tex]. Let's also suppose I know that 2 is an upper bound for [tex] S_n[/tex]. Then I want to show [tex] \forall \epsilon >0[/tex] [tex] \exists N[/tex] such that [tex] \forall n > N[/tex], [tex]| (1 + 1/(2n) )^n - \sqrt{e} | < \epsilon[/tex].

Does this argument work?

Lemma: [tex] \forall \epsilon > 0[/tex] [tex]\exists N_0[/tex] such that [tex] \forall n > N_0 [/tex], [tex] |2 + 1/n - 2| < \epsilon [/tex]. That is, [tex] lim_{n\rightarrow +\infty} {(2 + 1/n)} = 2[/tex].

[tex] \forall n | ( 1 + 1/(2n) )^n - \sqrt{e} | < |1/n|[/tex]. Since [tex] \forall n, 1/n > 0[/tex], then [tex] |1/n| = 1/n[/tex]. By our lemma, we know [tex] |1/n| = 1/n < \epsilon[/tex]. Choose an [tex] n > 1/ \epsilon [/tex]. Then [tex] | ( 1 + 1/(2n) )^n - \sqrt{e} | < \epsilon [/tex] and therefore [tex] lim_{n\rightarrow +\infty} { (1 + 1/(2n))^n } = \sqrt{e}[/tex].

I was originally trying to use * with the definition of a limit to show [tex] | ( 1 + 1/(2n) )^n - \sqrt{e} | < | ( 1 + 1/(2n) )^n - e | < \epsilon [/tex] but I couldn't figure out how to determine N.
 
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  • #2
It's not clear to me what the relevance of the lemma is to picking your value of n (why can't you just pick n larger than one over epsilon without looking at a different limit?), or why [tex]
\forall n | ( 1 + 1/(2n) )^n - \sqrt{e} | < |1/n|
[/tex] is true.
[tex]
| ( 1 + 1/(2n) )^n - \sqrt{e} | < | ( 1 + 1/(2n) )^n - e | < \epsilon
[/tex]

This middle term in the inequality is going to fairly large (in the analysis sense) when n is large. We know [tex]( 1 + 1/(2n) )^n[/tex] is going to be close to [tex] \sqrt{e}[/tex], so it can't be close to e as well. However, at some point you are going to need to use the limit that gives e, or some alternative definition of e
 
  • #3
It's not clear to me what the relevance of the lemma is to picking your value of n (why can't you just pick n larger than one over epsilon without looking at a different limit?), or why [tex] \forall n | ( 1 + 1/(2n) )^n - \sqrt{e} | < |1/n|[/tex] is true.

I added the lemma to use the fact I knew the limit of the 2+1/n was 2, and to bound the difference [tex]|(1 + 1/(2n))^n - \sqrt{e}| < |1/n|[/tex] for all n (n is of course restricted to naturals, and this doesn't include zero). So my reasoning was after establishing the above inequality, to use the lemma to argue I can find an n>N such that [tex]|1/n| = 1/n < \epsilon[/tex]. That such n being any integer [tex] > 1/\epsilon[/tex]. Therefore if [tex] n > 1/\epsilon[/tex], then [tex]|(1 + 1/(2n))^n - \sqrt{e}| < \epsilon[/tex]. Does this make sense?

This middle term in the inequality is going to fairly large (in the analysis sense) when n is large. We know [tex]( 1 + 1/(2n) )^n [/tex] is going to be close to [tex]\sqrt{e}[/tex] , so it can't be close to e as well. However, at some point you are going to need to use the limit that gives e, or some alternative definition of e

Sorry, I meant to say: [tex] |(1 + 1/(2n)^n - \sqrt{e}|[/tex] < [tex]|(1+1/n)^n - e|[/tex]. Then using the fact I that I knew [tex] \forall \epsilon[/tex] there was some N such that whenever n>N, I was trying to argue [tex]|(1+1/n)^n - e| < \epsilon[/tex]. The main problem I had was deriving an explicit N from the inequality above. That is why I choose to bound the difference [tex] |(1 + 1/(2n)^n - \sqrt{e}|[/tex] by [tex]1/n[/tex] instead.
 
Last edited:
  • #4
Here is an idea that I just thought of using the fact that I may assume * [tex] \lim_{n/rightarrow +/infty} {(1+1/n)^n}=e[/tex]. I am not sure if it is considered "rigorous" enough. Here it is:

I may use the fact that ** if [tex]S_n, T_n[/tex] are sequences, and [tex] \lim_{n\rightarrow +\infty} {S_n} =A[/tex], [tex] \lim_{n\rightarrow} {T_n} = B[/tex]. Then [tex] \lim_{n\rightarrow} {S_n T_n} = AB[/tex].

Suppose * and let m=2n. Then [tex] \lim_{n\rightarrow +\infty} {(1 + 1/n)^n} = \lim_{m\rightarrow +\infty} {(1+1/(2m))^{2m}[/tex] (Here I am making the argument that as [tex] n\rightarrow +\infty[/tex], [tex] 2m\rightarrow +\infty[/tex] and hence [tex] m\rightarrow +\infty[/tex]). By **, [tex] \lim_{n\rightarrow +\infty} {(1+1/(2m)^m (1+1/(2m))^m } = \lim_{m\rightarrow +\infty} {(1+1/(2m))^m} \lim_{m\rightarrow +\infty} {(1+1/(2m))^m = e[/tex]. Then [tex] \lim_{n\rightarrow +\infty} {(1+1/(2m))^m } = \sqrt{e} [/tex].
 

1. What is a proof of limit?

A proof of limit is a mathematical concept used to determine the behavior of a function as its input approaches a certain value. It is used to formally show that a function approaches a specific value as its input gets closer and closer to a given point.

2. How do I know if my proof of limit is correct?

A correct proof of limit should clearly state the function being evaluated, the limit value being approached, and the reasoning and steps used to show that the limit exists. It should also follow the proper mathematical notation and use logical reasoning and mathematical rules to support the conclusion.

3. What are some common mistakes when writing a proof of limit?

Some common mistakes include not clearly stating the function and limit value, using incorrect notation, skipping steps in the proof, and using incorrect mathematical rules or reasoning. It is important to be thorough and precise when writing a proof of limit to avoid these mistakes.

4. Can I use different methods to prove a limit?

Yes, there are multiple methods that can be used to prove a limit, such as the epsilon-delta method, the squeeze theorem, and direct substitution. The method chosen may depend on the function being evaluated and personal preference, but it is important to use a valid method and clearly explain the reasoning behind it.

5. How can I improve my skills in writing proofs of limits?

Practice is key in improving your skills in writing proofs of limits. It is also helpful to seek guidance from a mentor or teacher, and to study and understand the different methods and rules used in limit proofs. Additionally, reviewing and analyzing sample proofs can also help in developing your skills.

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