A question regarding the definition of e

[SafiBTA]

Homework Statement

In writing the definition of $e$ i.e. $e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have anything to do with positive integers (which we normally use 'n' for)?

Homework Equations

$e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$

The Attempt at a Solution

I see no clue as to why is this so.

 

[jbriggs444]

The n in that expression for e is not a free variable. It is a bound variable. It makes no sense to talk about the formula holding for some particular value of n. The n in the formula has a limited scope and is not defined at all outside of that scope.

Have you ever encountered a formal definition for "limit"? In your coursework, have you encountered statements such as "there exists an x in R such that for all n in N, ..."

 

[jedishrfu]

The limit describes a sequence of numbers that as n increases becomes closer and closer to e.

n=1 and the expr=2
n=2 and the expr=2.2
n=3 and the expr=2.37
n=4 and the expr=2.44
...
n approaches infinity expr approaches e

 

[jbriggs444]

To get a bit more pedantic, the "limit" is the single, fixed numeric value (if one exists) that the sequence approaches.

 

[Mark44]

In writing the definition of $e$ i.e. $e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have anything to do with positive integers (which we normally use 'n' for)?

By "formula" I presume you mean this expression: (1 + 1/n)ⁿ. Clearly it is not defined for n = 0. The "notation," as you called it, is a limit, and the expression whose limit is being taken has different values for different values of n (integer values). As n gets larger, the value of the expression (1 + 1/n)ⁿ gets closer to the number e.

 

[Ray Vickson]

Homework Statement

In writing the definition of $e$ i.e. $e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have anything to do with positive integers (which we normally use 'n' for)?

Homework Equations

$e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$

The Attempt at a Solution

I see no clue as to why is this so.

It is true that
$$e = \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x$$
but that is a theorem, rather than a definition.

The point is that concepts/objects need to be defined before they are used (in a strictly logical approach, at least), so that although one can define and understand $a^n$ for real $a$ and integer (or even rational) $n$, $a^x$ for real $x$ is trickier and more involved to get at. Sometimes, the definitions related to $a^x$ proceed through the use of the number $e$ itself, where it is understood that this $e$ is the one obtained by the usual integer limit.

 

[gianeshwar]

The limit describes a sequence of numbers that as n increases becomes closer and closer to e.

n=1 and the expr=2
n=2 and the expr=2.2
n=3 and the expr=2.37
n=4 and the expr=2.44
...
n approaches infinity expr approaches

Suppose there is a bank A which gives 100%interest per annum , and bank Bgives 100 percent but compounded twice in a year,and bank C gives 100 percent per annum but compounded thrice(i,e every for months ) and so on...
Now Dollar 1 deposited in each of these banks will be 2,(1plus1/2)power2,(1plus1/3) power3 and so on .
Ultimately in this way a hypothetical bank which compounds interest every moment will give you back Dollars e!