(1-1/n^2)^n Converge

5 min read Jun 16, 2024

The Limit of (1 - 1/n^2)^n: A Journey to e

The sequence (1 - 1/n^2)^n, as n approaches infinity, is a fascinating example in calculus that demonstrates the power of limits and how seemingly complex expressions can converge to a beautiful constant. Let's explore this sequence and understand why it converges to the mathematical constant e.

Understanding the Sequence

At first glance, (1 - 1/n^2)^n might seem intimidating. But let's break it down:

The base: (1 - 1/n^2) is a fraction slightly less than 1. As n gets larger, this fraction gets closer and closer to 1.
The exponent: n increases without bound.

The interplay between a base approaching 1 and an exponent growing infinitely large is crucial to the sequence's convergence.

The Limit: A Tale of Exponential Decay

As n grows, the base (1 - 1/n^2) approaches 1, but the exponent n also increases. This creates a tension between two competing forces:

The base approaching 1: This tends to make the entire expression approach 1 as well.
The exponent increasing: This tends to make the expression grow larger.

The key lies in how these forces interact. The base approaches 1 at a faster rate than the exponent increases, leading to a gradual decay in the value of the entire expression.

The Convergence to e

Using calculus techniques, we can rigorously prove that the limit of (1 - 1/n^2)^n as n approaches infinity is e, the base of the natural logarithm. This means that as n gets larger and larger, the value of the sequence gets closer and closer to e.

Here's a simplified explanation:

Rewrite the expression: We can rewrite (1 - 1/n^2)^n as [(1 + 1/n)(1 - 1/n)]^n.
Expand the expression: Expanding this expression using binomial theorem, we get a series with terms that approach the terms of the infinite series for e.
Take the limit: As n approaches infinity, the terms of the expanded series become increasingly similar to the terms of the infinite series for e, leading to the convergence to e.

Significance of the Limit

The convergence of (1 - 1/n^2)^n to e is significant because it connects seemingly disparate mathematical concepts:

Limits: The concept of a limit is fundamental to calculus and allows us to understand the behavior of functions as their inputs approach specific values.
Exponential functions: The constant e is the base of the natural logarithm and appears in many important mathematical formulas, particularly in the realm of exponential growth and decay.

The study of this sequence provides a fascinating glimpse into the intricate relationship between limits, exponential functions, and the fundamental constant e.