# The Number e .... Sohrab Proposition 2.3.15 ....

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In summary, Houshang H. Sohrab's book "Basic Real Analysis" discusses Chapter 2: Sequences and Series of Real Numbers. The conversation focuses on Proposition 2.3.15 and its proof, which involves showing that 1 / n! \leq 1 / (2 \cdot 3^{ n - 2 } ) for all n \geq 3. This is proven by recognizing that n! is the product of n factors, each of which is at least equal to 3, and taking the inverse. The conversation concludes with the grateful acknowledgement of assistance from castor28.
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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 an aspect of the proof of Proposition 2.3.15 ...

Proposition 2.3.15 and its proof read as follows:
View attachment 9052
In the above proof by Sohrab we read the following:

" ... ...Since $$\displaystyle 1 / n! \leq 1 / (2 \cdot 3^{ n - 2 } )$$ for all $$\displaystyle n \geq 3$$ ... ... "My question is ... how do we know this is true ... ?

Can someone please demonstrate how to prove that $$\displaystyle 1 / n! \leq 1 / (2 \cdot 3^{ n - 2 } )$$ for all $$\displaystyle n \geq 3$$ ... ...

Help will be much appreciated ...

Peter

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• Sohrab - Proposition 2.3.15 ... .png
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Hi Peter,

$n!=1\cdot2\cdot3\cdots n$ is the product of $n$ factors. As each of the $n-2$ factors after $2$ is at least equal to $3$, we have:
$$n!\ge1\cdot2\cdot3^{n-2}=2\cdot3^{n-2}$$
and the result follows by taking the inverse.

castor28 said:
Hi Peter,

$n!=1\cdot2\cdot3\cdots n$ is the product of $n$ factors. As each of the $n-2$ factors after $2$ is at least equal to $3$, we have:
$$n!\ge1\cdot2\cdot3^{n-2}=2\cdot3^{n-2}$$
and the result follows by taking the inverse.

Hmmm ... easy when you see how ... :) ...

Thanks for the help castor28 ...

Peter

## 1. What is "The Number e" and why is it important in mathematics?

"The Number e", also known as Euler's number, is a mathematical constant that is approximately equal to 2.71828. It is important in mathematics because it appears in many natural and scientific phenomena, such as compound interest, population growth, and radioactive decay. It is also a key component in the study of calculus and complex numbers.

## 2. Who discovered "The Number e" and when?

"The Number e" was discovered by Swiss mathematician Leonhard Euler in the 18th century. However, its existence and significance had been noted by earlier mathematicians such as John Napier and Jacob Bernoulli.

## 3. How is "The Number e" calculated?

"The Number e" can be calculated in several ways, including using the infinite series formula e = 1 + 1/1! + 1/2! + 1/3! + ..., or by taking the limit of (1 + 1/n)^n as n approaches infinity. It can also be found using a scientific calculator or computer program.

## 4. What is the relationship between "The Number e" and the natural logarithm?

The natural logarithm, denoted as ln(x), is the inverse function of e^x. This means that if e^x = y, then ln(y) = x. The natural logarithm is used to solve exponential equations involving "The Number e" and is also used in many applications such as finance and growth models.

## 5. How is "The Number e" used in real-world applications?

"The Number e" has many practical applications in fields such as finance, physics, and engineering. It is used to calculate compound interest, model population growth, and analyze electrical circuits. It is also used in statistics to calculate probabilities and in calculus to solve differential equations.

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