The Number e .... Another Issue/Problem Regarding Sohrab Proposition 2.3.15 ....

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In summary, Peter is trying to figure out how the statement "s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n) ..." leads to the statement that "t_n\geq 1+1+\ldots".
  • #1
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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with another issue/problem with the proof of Proposition 2.3.15 ...Proposition 2.3.15 and its proof read as follows:
View attachment 9076
In the above proof by Sohrab we read the following:

" ... ...Next, for any fixed \(\displaystyle m \in \mathbb{N}\) and \(\displaystyle n \geq m\), we have

\(\displaystyle t_n \geq 1 + 1 + \frac{1}{ 2! } \left( 1 - \frac{1}{ n } \right) + \ ... \ + \frac{1}{ m! } \left( 1 - \frac{1}{ n } \right) \ ... \ \left( 1 - \frac{m - 1}{ n } \right) \)so that letting \(\displaystyle n \to \infty\) we get

\(\displaystyle s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)\) ... ... "
My question is as follows:

How does the statement:" ... ... for any fixed \(\displaystyle m \in \mathbb{N}\) and \(\displaystyle n \geq m\), we have

\(\displaystyle t_n \geq 1 + 1 + \frac{1}{ 2! } \left( 1 - \frac{1}{ n } \right) + \ ... \ + \frac{1}{ m! } \left( 1 - \frac{1}{ n } \right) \ ... \ \left( 1 - \frac{m - 1}{ n } \right) \) ... " and letting \(\displaystyle n \to \infty\) ...
... ... lead to the statement that ... \(\displaystyle s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)\) ... ... In other words ... can someone explain in some detail how/why \(\displaystyle s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)\) ... ...is true?

Help will be appreciated ...

Peter
 

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  • #2
Hi Peter,

Peter said:
can someone explain in some detail how/why \(\displaystyle s_m = \sum_{ k = 0 }^m \frac{1}{ k! } \leq \text{ lim inf } (t_n)\) ... ...is true?

Take liminf on both sides of the inequality $$t_{n}\geq 1+1+\ldots$$ with respect to $n$. Since the limit of the right side with respect to $n$ exists and is equal to $s_{m}$, this is the value of the liminf of the right side.
 
  • #3
GJA said:
Hi Peter,
Take liminf on both sides of the inequality $$t_{n}\geq 1+1+\ldots$$ with respect to $n$. Since the limit of the right side with respect to $n$ exists and is equal to $s_{m}$, this is the value of the liminf of the right side.
Thanks GJA ...

Working on what you have suggested ...

Peter
 

FAQ: The Number e .... Another Issue/Problem Regarding Sohrab Proposition 2.3.15 ....

1. What is the number e and why is it important in mathematics?

The number e is a mathematical constant approximately equal to 2.71828. It is important because it appears in many different areas of mathematics, including calculus, differential equations, and complex analysis. It also has many real-world applications, such as in finance, physics, and biology.

2. How is the number e calculated?

The number e can be calculated in several ways, but one common method is by using the infinite series: e = 1 + 1/1! + 1/2! + 1/3! + ... where n! represents the factorial of n (n! = n x (n-1) x (n-2) x ... x 2 x 1).

3. What is Sohrab Proposition 2.3.15 and why is it important?

Sohrab Proposition 2.3.15 is a mathematical theorem that states that the limit of (1 + 1/n)^n as n approaches infinity is equal to the number e. This theorem is important because it provides a way to approximate the value of e and also has many applications in calculus and other areas of mathematics.

4. Can the number e be expressed as a finite decimal or fraction?

No, the number e cannot be expressed as a finite decimal or fraction. It is an irrational number, meaning it cannot be written as a ratio of two integers. Its decimal representation is infinite and non-repeating.

5. What are some real-world applications of the number e?

The number e has many real-world applications, including compound interest and continuous growth and decay problems in finance, population growth in biology, and electrical circuits in physics. It is also used in statistics, probability, and other areas of mathematics.

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