Problem 72 - 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. $$f(x)=x^{2} e^{x}$$

Step-by-Step Solution

Answer

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

Step 1: Write the power series for the exponential function

We are given the following power series for the exponential function \(e^x\): $$e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$$

Step 2: Multiply by \(x^2\)

To find the power series for \(f(x) = x^2 e^x\), we need to multiply the given power series of \(e^x\) by \(x^2\). Doing so, we obtain: $$f(x) = x^2 e^x = x^2 \sum_{k=0}^{\infty} \frac{x^k}{k!}$$

Step 3: Distribute the \(x^2\) into the sum

Now, we distribute the \(x^2\) into the sum: $$f(x) = \sum_{k=0}^{\infty} x^2 \frac{x^k}{k!} = \sum_{k=0}^{\infty} \frac{x^{k+2}}{k!}$$

Step 4: Rewrite the power series for \(f(x)\)

We can rewrite the power series of \(f(x)\) as follows: $$f(x) = \sum_{k=0}^{\infty} \frac{x^{k+2}}{k!} = \sum_{k=0}^{\infty} \frac{x^{k}}{(k-2)!}$$

Step 5: Determine the interval of convergence

Since the original power series for the exponential function converges for all \(x\), i.e., \(-\infty < x < \infty\), the power series for \(f(x) = x^2 e^x\) will also converge for the same interval. Thus, the interval of convergence for the resulting series is \(-\infty < x < \infty\). In conclusion, the power series representation for the function \(f(x) = x^2 e^x\) is $$f(x) = \sum_{k=0}^{\infty} \frac{x^{k}}{(k-2)!}$$ and the interval of convergence for the resulting series is \(-\infty < x < \infty\).

Key Concepts

Exponential Function

The exponential function, denoted as \( e^x \), is one of the most essential functions in mathematics, particularly due to its ubiquitous presence in calculus, complex numbers, and real-world applications. It's defined through an infinite power series:

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

This series sums an infinite number of terms where each term is determined by \( x^k/k! \)—a division of \( x \) raised to the power of \( k \), by \( k \) factorial.
The exponential function grows faster than all polynomial functions, but slower than functions like \( x^x \). A remarkable property of \( e^x \) is that it is equal to its own derivative. This makes it invaluable in differential equations, where it often models exponential growth or decay processes, such as population growth or radioactive decay.
Its power series converges for all real numbers, implying amazing predictability and consistency regardless of the value of \( x \). This infinite yet perfectly convergent nature of \( e^x \) plays a crucial role in mathematics and physics alike.

Interval of Convergence

The interval of convergence for a power series defines the set of values for which the series converges to a finite number. For any power series given by \( \sum_{k=0}^{\infty} a_k (x-c)^k \), determining this interval involves:

• Finding the values of \( x \) for which the series does not diverge.
• Checking the end points separately, as convergence can behave differently at these values.

For the exponential function \( e^x \), the power series \[ \sum_{k=0}^{\infty} \frac{x^k}{k!} \] converges for all values of \( x \).
This is because the factorial in the denominator grows much faster than the exponential function in the numerator, ensuring that each term of the series becomes very small, contributing to the overall convergence.
In the given exercise, where the goal is to find the series for \( f(x) = x^2 e^x \), we identify that the function still involves the series of \( e^x \). Thus, the interval of convergence remains unchanged, spanning all real numbers from \(-\infty\) to \(\infty\).

Series Convergence

A series is said to converge when its sequence of partial sums approaches a finite limit. For power series, convergence indicates that you can approximate any point in the interval of convergence with a finite number of terms, yielding a close approximation to the actual function value.

• For determining convergence, tools like the Ratio Test are often used.
• For exponential functions, \( e^x \), convergence is guaranteed across all \( x \).

Convergence of the series is critical in applications such as numerical computations and solving differential equations. For instance, even if computing tools only use a finite sum of an infinite series, this convergence ensures the result will closely match the actual function value.
In practice, convergence allows engineers and scientists to compute results much faster than evaluating exact solutions, making it possible to handle otherwise computationally expensive problems. The series representation of functions, thanks to convergence, enables smooth transitions between analytic calculations and practical applications in technology.