Problem 75 - Applied Calculus

Question

Find, rounding to five decimal places: a. \(\left(1+\frac{1}{100}\right)^{100}\) b. \(\left(1+\frac{1}{10,000}\right)^{10,000}\) c. \(\left(1+\frac{1}{1,000,000}\right)^{1,000,000}\) d. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4

Step-by-Step Solution

Answer

The limiting value of the sequence is approximately 2.71828, which is the mathematical constant \(e\).

Step 1: Calculate Part (a)

To solve the problem \( \left(1+\frac{1}{100}\right)^{100} \), we need to perform the exponentiation using the base of \(1+\frac{1}{100}\). First, compute \(1+\frac{1}{100} = 1.01\). Raise this to the 100th power: \(1.01^{100} \approx 2.70481\).

Step 2: Calculate Part (b)

Now solve \( \left(1+\frac{1}{10,000}\right)^{10,000} \). First, calculate \(1+\frac{1}{10,000} = 1.0001\). Then, raise this to the power of 10,000: \(1.0001^{10,000} \approx 2.71815\).

Step 3: Calculate Part (c)

Next, solve \( \left(1+\frac{1}{1,000,000}\right)^{1,000,000} \). First, compute \(1+\frac{1}{1,000,000} = 1.000001\). Then perform the exponentiation: \(1.000001^{1,000,000} \approx 2.71828\).

Step 4: Determine Limiting Value

Look at the results from parts (a), (b), and (c). As the number in the exponent grows larger and the fraction gets smaller, the results converge toward a specific number. From the calculations, this number appears to be approximately 2.71828.

Key Concepts

Exponential Calculations

Exponential calculations involve raising a number, known as the base, to a certain power, which is called the exponent. In this exercise, we frequently encounter expressions such as \[ \left(1 + \frac{1}{n}\right)^{n} \]. The base in this case is \(1 + \frac{1}{n}\), and it is raised to the power \(n\). This type of calculation is crucial in understanding how quantities grow over time, especially in fields like finance and natural sciences.
- Start by calculating the base: add the fraction to 1.- Then, raise this base to the specified power (the exponent).
As you perform these calculations, remember that exponentiation can be thought of as repeated multiplication. For example, \(1.01^{100}\) means multiplying 1.01 by itself 100 times. Advanced calculators or software can help handle large exponentiations, as they become impractical to compute manually.

Convergence

Convergence is a crucial concept in calculus, and it describes how a sequence approaches a specific value as the terms of the sequence increase. In the case of exponential calculations like those in our exercise, we observe convergence when the results of the calculations stabilize around a particular number.
For example, when you examined \[ \left(1 + \frac{1}{n}\right)^{n} \], with increasing values of \(n\), you'll see the outcome tends to approach a certain value. Initially, with smaller values of \(n\), the results may vary significantly, such as the 2.70481 found in part (a).
However, as more calculations are made using larger values of \(n\), as in parts (b) and (c), the sequence's results start to stabilize, pointing towards a specific numeric value. Recognizing convergence is key to understanding limits, a fundamental aspect of calculus.

Natural Number e

The natural number \( e \) is one of the most important constants in mathematics. It is approximately equal to 2.71828, and it surfaces naturally in exponential growth and decay processes, as well as in calculus.
In the exercise, when you look at expressions like \[ \left(1 + \frac{1}{n}\right)^{n} \] with increasingly larger \(n\), they converge to this number \( e \). This constant is vital in defining functions like the natural exponential function \( e^x \), which models continuous growth.
- \( e \) is a transcendental number, meaning it's not a solution to any non-zero polynomial equation with rational coefficients.- It is frequently used in formulas involving compound interest, differential equations, and statistics.
Understanding \( e \) and its properties allows you to analyze a variety of real-world phenomena, such as population growth, resource consumption, and financial investments.