(1+2/n)^n

4 min read Jun 16, 2024

The Limit of (1 + 2/n)^n as n Approaches Infinity

The expression (1 + 2/n)^n is a fascinating one in mathematics, as it demonstrates a fundamental concept in calculus: the limit of a sequence. As n grows increasingly large, this expression approaches a specific value, revealing a connection to the exponential function e. Let's explore this further.

Understanding the Expression

At its core, this expression represents a sequence. As n takes on consecutive natural numbers (1, 2, 3, ...), the value of the expression changes. For small values of n, the expression is relatively simple to calculate. For instance:

When n = 1, the expression equals (1 + 2/1)^1 = 3
When n = 2, the expression equals (1 + 2/2)^2 = 4

However, as n gets larger, the calculation becomes more complex. The question arises: what happens to this expression as n approaches infinity?

The Limit and its Connection to 'e'

It turns out that as n approaches infinity, the expression (1 + 2/n)^n converges to a specific value: e^2, where e is Euler's number, approximately equal to 2.71828. This is a fundamental result in calculus and has profound implications in various fields.

To understand why this happens, we need to consider the following:

The role of the exponent: As n increases, the exponent grows, causing the expression to increase rapidly.
The role of the base: The base (1 + 2/n) gets increasingly closer to 1 as n approaches infinity.

This interplay between the increasing exponent and the base approaching 1 leads to a unique limit.

Applications

The limit of (1 + 2/n)^n has wide-ranging applications, including:

Compound Interest: The expression directly relates to the concept of compound interest, where interest is added to the principal periodically. As the frequency of compounding increases (n approaches infinity), the final amount approaches the value of e^2 times the principal.
Exponential Growth: The limit arises in models of exponential growth, such as population growth or the spread of a disease.
Calculus: This expression plays a crucial role in the development of calculus, particularly in the definition of the exponential function and its derivative.

Conclusion

The expression (1 + 2/n)^n offers a glimpse into the power of mathematical limits and their connection to fundamental constants like e. As n approaches infinity, this expression converges to e^2, demonstrating the profound impact of this limit in various scientific and mathematical fields. Understanding the behavior of this expression provides insights into concepts like exponential growth, compound interest, and the fundamental nature of calculus.