Proving the Limit and Definition of Euler's Number (e)

Understanding the concept of Euler's number, denoted as (e), is crucial for many areas of mathematics and its applications in science and engineering. One of the most elegant ways to define (e) is through the limit of a sequence. This article will explore various methods to prove the limit (lim_{ntoinfty} left(1 frac{1}{n}right)^n e). We will also discuss the connection between logarithms, Taylor series, and the definition of (e).

The Limit and Its Form

The limit (lim_{ntoinfty} left(1 frac{1}{n}right)^n) is an indeterminate form of type (1^infty). To resolve this, we often take the natural logarithm of the expression and then evaluate the limit of the resulting expression.

Using the Natural Logarithm Method

Let (L lim_{ntoinfty} left(1 frac{1}{n}right)^n). We then use the natural logarithm to transform the limit:

(ln L lim_{ntoinfty} n ln left(1 frac{1}{n}right))

As (n to infty), (frac{1}{n} to 0), and thus the limit (lim_{xto 0} frac{ln (1 x)}{x} 1). This can be shown using the series expansion for (ln (1 x)).

Alternatively, by substituting (m frac{1}{n}) as (m to 0), we get:

(lim_{mto 0} frac{1}{m} ln (1 m))

Expanding (ln (1 m)) using the Taylor series for (ln (1 x)) around (x 0):

(ln (1 m) approx m - frac{m^2}{2} frac{m^3}{3} - cdots)

Thus, we have:

(lim_{mto 0} frac{1}{m} left(m - frac{m^2}{2} frac{m^3}{3} - cdotsright) 1)

Therefore, (ln L 1) and hence (L e).

Another Approach Using L'H?pital's Rule

We can also approach the limit using L'H?pital's rule. For this, we rewrite the limit in a different form:

(L lim_{ntoinfty} left(1 frac{1}{n}right)^n)

Take the natural logarithm:

(ln L lim_{ntoinfty} n ln left(1 frac{1}{n}right))

This is in an indeterminate form (frac{0}{0}), so we use L'H?pital's rule:

(ln L lim_{ntoinfty} frac{ln left(1 frac{1}{n}right)}{frac{1}{n}} lim_{ntoinfty} frac{frac{1}{1 frac{1}{n}} cdot left(-frac{1}{n^2}right)}{-frac{1}{n^2}} lim_{ntoinfty} frac{1}{1 frac{1}{n}} 1)

The natural logarithm of (L) is 1, hence (L e).

Definition and Properties of Euler's Number

The number (e) is defined as:

(e lim_{ntoinfty} left(1 frac{1}{n}right)^n)

The definition is equivalent to proving that:

(lim_{xto 0} frac{ln (1 x)}{x} 1)

This can be seen using the series expansion of (ln (1 x)) around (x 0).

We can also consider the sequence generated by a change of variable and integral representation. Specifically:

(lim_{hto 0} frac{log (x^h - log x)}{h} frac{1}{x})

For a sequence (h frac{1}{2}, frac{1}{3}, ldots, frac{1}{n}, ldots), we get:

(A lim_{ntoinfty} n cdot log left(frac{x^{1/n}}{x}right) log left(frac{1}{x}^nright))

Summing up:

(log left(frac{1}{x}^nright) to frac{1}{x} quad text{for} quad n to infty)

Changing the variable to (y frac{1}{x}), we obtain:

(lim_{ntoinfty} left(1 frac{y}{n}right)^n e^y)

In the special case where (y 1):

(e lim_{ntoinfty} left(1 frac{1}{n}right)^n)

Conclusion

The limit (lim_{ntoinfty} left(1 frac{1}{n}right)^n) can be proven through several methods, including the use of natural logarithms, L'H?pital's rule, and integral representations. These techniques highlight the elegance and the deep mathematical nature of the constant (e), which is fundamental in many areas of mathematics and its applications.