Very simple question about limit superior (or upper limit)

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This is because limsup is a limit of subsequences, and there is a subsequence of Sn which converges to e. In order to show that limsup Sn < e, you would need to show that there is no subsequence converging to e, which is not true.
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jwqwerty
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Homework Statement



If Sn =1/0! + 1/1! + 1/2! +... 1/n! , Tn=(1+(1/n))^n, then lim sup Tn ≤ e? (e=2.71...)

Homework Equations



1. e= Ʃ(1/n!)
2. If Sn≤Tn for n≥N, then lim sup Sn ≤ lim sup Tn

The Attempt at a Solution



By binomial theorem,
Tn= 1 + 1 + 1/(2!)(1-1/n) + 1/(3!)(1-1/n)(1-2/n) + ... + 1/n!
Hence Tn ≤ Sn< e,
lim sup Tn ≤ lim sup Sn< e

∴ lim sup Tn< e

But I do not get lim sup Tn ≤ e

What did i do wrong?
 
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  • #2
1. If you show that limsup Tn < e, then it follows that limsup Tn ≤ e. It turns out that it is not true that limsup Tn < e (so you did mess up in coming to that conclusion), but this conclusion does not contradict the claim that Tn ≤ e.
2. If Tn ≤ Sn < e, then we can conclude that limsup Tn ≤ limsup Sn ≤ e. We cannot, however, conclude that limsup Sn < e based on this information; in fact, limsup Sn = e.
 

Related to Very simple question about limit superior (or upper limit)

1. What is the definition of limit superior (or upper limit)?

Limit superior, also known as upper limit or supremum, is a mathematical concept that describes the largest possible limit of a sequence or function as its index approaches infinity.

2. How is limit superior different from limit inferior?

While limit superior represents the largest possible limit of a sequence or function, limit inferior represents the smallest possible limit. In other words, limit superior is the highest possible bound, while limit inferior is the lowest possible bound.

3. How is limit superior calculated?

To calculate limit superior, one must first take the supremum (or highest value) of the set of all possible limits of a sequence or function. This is done by taking the limit of the sequence or function as its index approaches infinity. If the supremum exists, it is equal to the limit superior.

4. What are some applications of limit superior in science?

Limit superior is widely used in various fields of science, such as physics, engineering, and computer science. It is especially useful in analyzing and predicting the behavior of systems that involve infinite sequences or functions, such as in the study of fractals and chaotic systems.

5. How does limit superior relate to convergence and divergence?

If the limit superior of a sequence or function is finite, it indicates that the sequence or function converges. On the other hand, if the limit superior is infinite, the sequence or function diverges. In other words, limit superior can be used to determine whether a sequence or function approaches a finite value or not.

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