Problem 83 - College Algebra

Question

As \(n\) increases, the terms of the sequence $$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$ get closer and closer to the number \(e\) (where \(e \approx 2.7183\) ). Use a calculator to find \(a_{10}, a_{100}, a_{1000}, a_{10}, a_{00},\) and \(a_{100,000}\) comparing these terms to your calculator's decimal approximation for \(e\).

Step-by-Step Solution

Answer

For \(n=10, 100, 1000, 10000, 100000\), the sequence \(a_{n}=\left(1+\frac{1}{n}\right)^{n}\) yields the values approximately 2.59374, 2.70481, 2.71692, 2.71815, 2.71827 respectively. As \(n\) increases, these values approach the value of the mathematical constant \(e\), approximately 2.7183.

Step 1: Understand the sequence

We are given a sequence \(a_{n}=\left(1+\frac{1}{n}\right)^{n}\). This sequence is a mathematical representation of the value of the mathematical constant \(e\).

Step 2: Write down the sequence for given values of n

We will write down the sequence for \(n=10, 100, 1000, 10000, 100000\) as follows: \[a_{10}=\left(1+\frac{1}{10}\right)^{10}\], \[a_{100}=\left(1+\frac{1}{100}\right)^{100}\], \[a_{1000}=\left(1+\frac{1}{1000}\right)^{1000}\], \[a_{10000}=\left(1+\frac{1}{10000}\right)^{10000}\], \[a_{100000}=\left(1+\frac{1}{100000}\right)^{100000}\]

Step 3: Calculate values

By inserting these values into a calculator, we get the following results: \(a_{10}\approx2.59374\), \(a_{100}\approx2.70481\), \(a_{1000}\approx2.71692\), \(a_{10000}\approx2.71815\), \(a_{100000}\approx2.71827\)

Step 4: Compare with e

When we compare these results with the value of \(e\) from a calculator (approximately 2.7183), we notice that as \(n\) increases, the value of \(a_n\) gets closer and closer to \(e\), which illustrates the concept that the given sequence is a representation of the mathematical constant \(e\).

Key Concepts

sequence and series

A sequence is an ordered list of numbers. In mathematics, a series is the sum of the terms of a sequence. Both concepts are foundational in understanding how values can converge or diverge over time. Let's take the sequence \(a_{n}=\left(1+\frac{1}{n}\right)^{n}\) as an example. In this sequence, each term is calculated based on its position, denoted as \(n\). For example, \(a_{10}\), \(a_{100}\) shows how each term develops as \(n\) increases.

• With sequences like \(a_n\), we observe how values grow or approach a certain limit, such as the mathematical constant \(e\).
• This specific sequence approaches \(e\) as \(n\) grows larger, demonstrating convergence.

This sequence gradually increases its values, coming closer to \(e\), as seen with \(n = 100000\), which gets extremely close to the value of 2.7183. This ordered progression towards \(e\) beautifully illustrates the concept of sequences and series in mathematics.

limits

Limits are a critical concept in calculus that tells us how a function behaves as its input approaches a certain value. In the context of the sequence \(a_{n}=\left(1+\frac{1}{n}\right)^{n}\), the limit is essential to understanding how the sequence progresses towards the mathematical constant \(e\).
By evaluating this limit, we learn that as \(n\) gets larger, the output of the sequence gets closer to \(e\). Let's break it down:

• As \(n\) approaches infinity, the fractions \(\frac{1}{n}\) become very small, making \(1 + \frac{1}{n}\) effectively become closer to 1.
• Raising this to the power of \(n\) causes \(a_n\) to approach the limit \(e\).

This limit behavior demonstrates how a sequence like \(a_n\) can approximate a constant. It provides a vivid demonstration of how limits help mathematicians and scientists understand the behavior of functions under various conditions.

mathematical constant

The mathematical constant \(e\) is one of the most important numbers in mathematics, akin to π in its ubiquity and usefulness. \(e\) is an irrational number, meaning it cannot be expressed as a simple fraction. Its decimal expansion goes on forever without repeating. \
Renowned for its critical role in calculus, particularly in situations involving growth and decay, \(e\) emerges naturally in various mathematical contexts. Here’s why \(e\) is special:

• It is approximately equal to 2.7183, as derived from sequences like \(a_{n}=\left(1+\frac{1}{n}\right)^{n}\).
• Appears in numerous formulas, such as the exponential growth or decay formula \(y = e^{x}\).
• Fundamental in defining natural logarithms, which are logarithms to the base \(e\), commonly used in sciences.

Whether studying compound interest in finance, population models in biology, or wave functions in physics, \(e\) plays a pivotal role in comprehending complex problems across disciplines.