Problem 71 - Calculus Early Transcendentals

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. $$f(x)=e^{-3 x}$$

Step-by-Step Solution

Answer

The power series representation for the function \(f(x) = e^{-3x}\) is given by: $$ f(x) = e^{-3x} = \sum_{k=0}^{\infty} \frac{(-3)^{k}x^{k}}{k!} $$ The interval of convergence for this power series is for all real numbers \(-\infty < x < \infty\).

Step 1: Write down the given power series for the exponential function

The power series for the exponential function centered at 0 is given as: $$ e^x = \sum_{k=0}^{\infty} \frac{x^{k}}{k!} $$

Step 2: Substitute \(-3x\) in place of \(x\) in the power series for \(f(x)\)

Now, we will substitute \(-3x\) in place of \(x\) in the given power series to find the power series for \(f(x) = e^{-3x}\): $$ f(x) = e^{-3x} = \sum_{k=0}^{\infty} \frac{(-3x)^{k}}{k!} $$

Step 3: Simplify the power series expression

Let's simplify the power series expression for \(f(x)\): $$ f(x) = e^{-3x} = \sum_{k=0}^{\infty} \frac{(-3)^{k}x^{k}}{k!} $$

Step 4: Find the interval of convergence

For the given power series of the exponential function, it converges for all real numbers (-∞ < x < ∞). Since \(f(x) = e^{-3x}\) is also an exponential function, its interval of convergence will also be for all real numbers: $$ -\infty < x < \infty $$ So, the power series for the function \(f(x) = e^{-3x}\) is given by: $$ f(x) = e^{-3x} = \sum_{k=0}^{\infty} \frac{(-3)^{k}x^{k}}{k!} $$ And it converges for all real numbers \(-\infty < x < \infty\).

Key Concepts

Power Series

A power series is a way to express a function as an infinite sum of terms involving powers of a variable. Imagine you have a function, and you want to express it as an infinite polynomial. That's a power series! Each term in this series has a coefficient and is raised to an increasing power of the variable.

For example, the power series for the exponential function is:

• \[ e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}\]

In this expression:

• \(x^k\) is the variable raised to the power \(k\), where \(k\) is an integer starting from 0.
• \(k!\) is the factorial of \(k\), which is the product of all positive integers up to \(k\). Factorials grow fast, making the terms smaller as \(k\) increases, which is important for convergence.

Using power series, we can approximate functions or solve differential equations, making them a powerful tool in calculus and analysis. By substituting values into the series, we can explore new functions, like turning \(e^x\) into something more complex through substitution.

Interval of Convergence

The interval of convergence is a crucial concept when dealing with power series. It determines the set of values for which the series converges, meaning it sums up to a finite number.

For many power series, there's a range of values (an interval) where the series behaves nicely, and outside this range, it might not converge at all. For example, the interval of convergence for the exponential function's power series is all real numbers:

• \[-\infty < x < \infty\]

This is because exponential functions grow smoothly and are well-behaved for all real numbers. When working with other functions derived from a power series like \(f(x) = e^{-3x}\), you apply similar analysis to find out if and when the series changes its behavior. Generally, exponential function power series tend to have the same interval because modifying the exponent doesn't affect convergence if the entire structure retains its form.

• Methods such as the Ratio Test or Root Test can determine convergence and find this interval.

Knowing the interval is essential for safely utilizing a power series to approximate functions or solve equations.

Infinite Series

An infinite series is essentially a sum of an infinite sequence of terms. Think of it as adding numbers endlessly. In mathematics, we use these to extend our understanding of sums beyond finite boundaries.

Key characteristics include:

• Each term in an infinite series contributes to a larger whole, carrying us closer to a specific value when it converges.
• Convergence means that as we add more and more terms, the series approaches a particular number, even as it continues infinitely.

Infinite series can be daunting because they literally never end, but tools like power series can help us simplify complex functions through their infinite nature. This allows for more manageable computation in real-world applications.

With our function \(f(x) = e^{-3x}\) series representation, we rely on the infinite series definition:

• \[ \sum_{k=0}^{\infty} \frac{(-3)^k x^k}{k!}\]

Despite its infinite nature, this arrangement utilizing factorals ensures convergence over the entire real line. Thus, infinite series are a vibrant area of calculus, offering tools that break beyond finite limitations.