Question

Exponential function In Section 3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty < x < \infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=e^{2 x}$$

Short answer

Answer: The power series representation for the function \(f(x) = e^{2x}\) is: $$e^{2x} = \sum_{k=0}^\infty \frac{2^kx^k}{k!}$$ The interval of convergence for this power series is \(-\infty < x < \infty\).

Step-by-step solution

Substitute 2x in the power series for e^x

The power series for the exponential function is given by: $$e^x =\sum_{k=0}^{\infty} \frac{x^k}{k!}$$ Now, we substitute \(2x\) in place of \(x\): $$e^{2x} = \sum_{k=0}^\infty \frac{(2x)^k}{k!}$$

Simplify the power series for e^(2x)

We can simplify the power series as follows: $$e^{2x} = \sum_{k=0}^\infty \frac{2^kx^k}{k!}$$ By factoring out \(2^k\), we have the power series representation for \(f(x) = e^{2x}\).

Find the interval of convergence

The interval of convergence for the exponential function \(e^x\) is \(-\infty < x < \infty\), as given. The power series for \(e^{2x}\) contains a substitution of \(2x\), which does not change the interval of convergence. Therefore, the interval of convergence for \(f(x) = e^{2x}\) is also \(-\infty < x < \infty\). In summary, the power series representation for the function \(f(x) = e^{2x}\) is: $$e^{2x} = \sum_{k=0}^\infty \frac{2^kx^k}{k!}$$ And its interval of convergence is \(-\infty < x < \infty\).

Key concepts

Understanding the Exponential Function

The exponential function is one of the most fundamental functions in mathematics. It is often denoted as \(e^x\), where \(e\) is approximately equal to 2.71828. This function is unique because it is its own derivative, meaning that when you take the derivative of \(e^x\), it remains \(e^x\).

This property makes it immensely valuable in calculus and other fields of advanced mathematics. The exponential function is used in various real-world applications, including modeling population growth, compound interest, and radioactive decay.

• Exponential Growth: Describes how quantities grow at a rate proportional to their current amount.
• Natural Exponential Function: \(e^x\) serves as the base for natural exponential processes.
• Derivative: The derivative of \(e^x\) is itself, i.e., \(\frac{d}{dx}e^x = e^x\).

When working with the exponential function, we often turn to its power series representation, which helps in analyzing it in calculus, particularly when solving differential equations. The power series is an infinite sum that converges to \(e^x\) for any real number \(x\).

Overall, the exponential function plays a pivotal role in mathematical modeling and problem-solving across multiple disciplines.

Interval of Convergence Explained

When dealing with power series, it's important to determine where the series converges. The interval of convergence is the set of all \(x\) values for which a power series converges to a finite limit. For most standard series, this can be found using techniques like the ratio or root tests.

For example, the power series of the exponential function \(e^x\): \[ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} \] converges for all real numbers \(x\), meaning its interval of convergence is \(-\infty < x < \infty\).

This is due to the fact that the exponential function grows faster than any polynomial, yet its series' terms decrease very rapidly due to the factorial in the denominator. Thus, when substituting \(2x\) into the power series for \(e^x\) to get \(e^{2x}\)\[ e^{2x} = \sum_{k=0}^\infty \frac{2^k x^k}{k!} \] the interval of convergence remains unchanged, still \(-\infty < x < \infty\).

Understanding the interval of convergence is crucial in calculus as it tells us where the series represents the function accurately within a given range of values.

Connections with Calculus

Calculus is the branch of mathematics that studies how things change. It provides tools to investigate rates of change (differentiation) and accumulations of quantities (integration). The exponential function and its power series representation are key components in both these operations.

In differentiation, we use the property that the derivative of \(e^x\) is itself. This characteristic is pivotal in solving differential equations, where it often emerges as a solution form due to its unique properties. When considering functions such as \(f(x) = e^{2x}\), calculus allows us to handle derivatives and integrals involving exponential terms with ease, due to their straightforward nature.

• Differential Equations: Exponential functions are common solutions in these equations.
• Chain Rule: Used for functions like \(e^{2x}\), enabling us to take derivatives efficiently, such as \(2e^{2x}\).

In integration, the exponential function helps express solutions to integrals that otherwise might be complex, especially those that model real-world scenarios like growth and decay processes.

Therefore, understanding the exponential function and its role in calculus enables students to apply mathematical concepts effectively across a variety of problems and disciplines.