Understanding the Limit of (1 1/n)^n as n Approaches Infinity: The Emergence of Eulers Number e

Understanding the Limit of (1 1/n)^n as n Approaches Infinity: The Emergence of Euler's Number e

In this article, we delve into the mathematical concept of the limit of ((1 frac{1}{n})^n) as (n) approaches infinity. This limit is a fundamental expression in calculus, leading to the discovery of the mathematical constant (e).

Introduction

The expression ((1 frac{1}{n})^n) has significant implications in various fields of mathematics and science. When (n) is infinite, this expression simplifies to Euler's number, denoted by (e). This article will explore the mathematical rigor and the historical context of this important constant.

The Expressions and Their Limit

Consider the expression (lim_{n to infty} (1 frac{1}{n})^n). By definition, as (n) approaches infinity, the term (frac{1}{n}) approaches zero, making the term inside the parentheses approach 1. However, despite this, the overall expression does not necessarily 'collapse' to 1. Instead, it reaches a specific, finite value which is the Euler's number, (e).

The Binomial Theorem Approach

To understand this better, we can use the Binomial Theorem, which states that ((ab)^n sum_{k0}^{n} {binom{n}{k} a^{n-k} b^k}).

Let (a 1) and (b frac{1}{n}). Then:

((1 frac{1}{n})^n sum_{k0}^{n} {binom{n}{k} (frac{1}{n})^k})

Expanding the sum, we get:

((1 frac{1}{n})^n 1 frac{n}{1!} left(frac{1}{n}right) frac{n(n-1)}{2!} left(frac{1}{n}right)^2 frac{n(n-1)(n-2)}{3!} left(frac{1}{n}right)^3 cdots)

Further simplifying, this becomes:

((1 frac{1}{n})^n 1 1 frac{(1-frac{1}{n})}{2!} frac{(1-frac{1}{n})(1-frac{2}{n})}{3!} cdots)

The Limit as n Approaches Infinity

Now, let's analyze the limit as (n) approaches infinity:

(lim_{n to infty} (1 frac{1}{n})^n lim_{n to infty} sum_{k0}^{n} frac{(1-frac{1}{n})(1-frac{2}{n})cdots(1-frac{k-1}{n})}{k!})

As (n) approaches infinity, the terms (frac{1}{n}), (frac{2}{n}), (frac{3}{n}), etc., all approach zero, thus simplifying the expression on the right-hand side to:

(sum_{k0}^{infty} frac{1}{k!})

This series is known as the Taylor series expansion of the exponential function (e^x) evaluated at (x 1), which is the definition of (e).

Euler's Number and Its Significance

The number (e) is approximately 2.71828 and is one of the most important numbers in mathematics. It appears in various contexts, from compound interest and exponential growth to complex analyses in physics and engineering. It is a cornerstone in the development of calculus and differential equations.

Historical Context

The discovery of (e) is often attributed to Jacob Bernoulli, who investigated the problem of compound interest. The actual term "Euler's Number" was not used until the 18th century and named after Leonhard Euler, who made significant contributions to this field of mathematics.

Final Proof

Thus, we can conclude:

(lim_{n to infty} (1 frac{1}{n})^n e)

Putting (n 1/m) and taking the limit as (m) approaches 0 (equivalent to (n) approaching infinity), we get:

(lim_{m to 0} (1 m)^{1/m} e)

This expression clearly shows that the limit of ((1 m)^{1/m}) as (m) approaches 0 is indeed the constant (e).