Problem 45 - Calculus Early Transcendentals

Question

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$e^{2}$$

Step-by-Step Solution

Answer

Answer: The first four nonzero terms of the infinite series representing the value of \(e^{2}\) are \(1, 2, 2,\) and \(\frac{4}{3}\).

Step 1: Recall the Taylor series for exponential function

The Taylor series for the exponential function is given by: $$e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

Step 2: Replace x with 2 in the Taylor series

To find the infinite series for \(e^{2}\), replace x with 2 in the Taylor series: $$e^{2} = \sum_{n=0}^{\infty} \frac{2^n}{n!} = 1 + 2 + \frac{2^2}{2!} + \frac{2^3}{3!} + \cdots$$

Step 3: Calculate the first four nonzero terms

Now, we'll calculate the first four nonzero terms of the series: 1. The first term is \(1\). 2. The second term is \(2\). 3. The third term is \(\frac{2^2}{2!} = \frac{4}{2} = 2\). 4. The fourth term is \(\frac{2^3}{3!} = \frac{8}{6} = \frac{4}{3}\). Thus, the first four nonzero terms of the infinite series equal to \(e^{2}\) are \(1, 2, 2,\) and \(\frac{4}{3}\). The series can be written as: $$e^{2} \approx 1 + 2 + 2 + \frac{4}{3} + \cdots$$

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a variable acts as the exponent. The most famous exponential function is the base of the natural logarithms, denoted as \( e^x \), where \( e \) is approximately equal to 2.71828. Exponential functions have a wide range of applications in mathematics, physics, and engineering, frequently used to model growth and decay processes.
The importance of the exponential function in series expansions is that it allows us to approximate its value using a series of easier computations. This approximation becomes handy when calculating powers of \( e \) like \( e^2 \), as seen in the Taylor series.

Infinite Series

Infinite series are sums that continue indefinitely, potentially having an infinite number of terms. They are crucial for understanding concepts in calculus and analysis.
The Taylor series is a type of infinite series used to represent functions like the exponential function by an infinite sum of terms calculated from the values of its derivatives at a single point. By converting complex functions into simpler infinite series forms, we can evaluate these functions for various input values, making calculations more accessible.

Nonzero Terms

When dealing with infinite series, especially in problems like calculating \( e^2 \) using a Taylor series, finding the first few nonzero terms gives an approximation of the function.
Nonzero terms are the meaningful parts of a series as they contribute to building an accurate approximation. In the example, the first four nonzero terms of \( e^2 \) calculated using the Taylor series are 1, 2, 2, and \( \frac{4}{3} \). Each term builds on the previous, getting the series closer to the actual value of the function.

Calculation Steps

The process of finding the first few terms in the Taylor series can be broken down into manageable steps:

• Start with the known Taylor series formula: For the exponential function, this is \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \).
• Substitute the desired value into the series: If calculating \( e^2 \), substitute \( x = 2 \).
• Compute individual terms: Identify and calculate the first few nonzero terms one at a time, such as \( 1, 2, 2, \) and \( \frac{4}{3} \).
• Add the terms for an approximation: Combine them to approximate \( e^2 \). The more terms calculated, the closer the approximation will be to \( e^2 \).

Breaking down the computation in this way makes complex calculations more approachable and accurate.