Question

A function \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is given. (a) Give a power series for \(f^{\prime}(x)\) and its interval of convergence. (b) Give a power series for \(\int f(x) d x\) and its interval of convergence. $$\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}$$

Short answer

(a) \( f'(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2n-1}}{(2n-1)!} \), interval: \([-\infty, \infty]\). (b) \( \int f(x) \, dx = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n+1}}{(2n+1)(2n)!} \), interval: \([-\infty, \infty]\).

Step-by-step solution

Identify the Function Type

The original function is given as a series: \( f(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \), which resembles the series expansion of \( \cos(x) \).

Express Function as a Series

The function \( f(x) \) can be expressed in series form, where it is already defined as \( \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \). Hence, we explicitly note this form as already given.

Compute the Derivative Series

To find the derivative of \( f(x) \), differentiate term-by-term: \( f'(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n} (2n) x^{2n-1}}{(2n)!} \). Simplify each term to \( \sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2n-1}}{(2n-1)!} \).

Determine Derivative Series Interval of Convergence

The interval of convergence for \( f(x) = \cos(x) \) is \([-\infty, \infty]\). Since taking the derivative doesn't change the radius of convergence, \( f'(x) \) also converges in \([-\infty, \infty]\).

Compute the Integral Series

To integrate \( f(x) \), integrate each term: \( \int f(x) \, dx = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n+1}}{(2n+1)(2n)!} \).

Determine Integral Series Interval of Convergence

Integrating a power series might change the interval of convergence at the endpoints. For \( \cos(x) \), which converges absolutely, the integrated series converges on \([-\infty, \infty]\) as well.

Key concepts

Convergence Interval

When dealing with power series, understanding the convergence interval is crucial. For a given power series, the interval of convergence is the range of values of the variable where the series converges to a specific value. This includes analyzing the radius of convergence and the behavior of endpoints. In our case, the given series resembles the Taylor series of the cosine function, \[ f(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \], which is known to converge for all real numbers. Thus, the interval is \[-\infty, \infty\]. This implies that regardless of the value of \( x \), the series will present a valid function of \( f(x) \).
Keep in mind that both differentiation and integration of the series will not alter its radius of convergence. However, they may affect the behavior at the endpoints.

Derivative of a Series

Calculating the derivative of a power series involves taking the derivative term by term. For the series \( f(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \), the differentiation process results in a new series:
\[ f'(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2n-1}}{(2n-1)!} \].
This transformation involves reducing the power of \( x \) in each term by one and multiplying by the previous power's coefficient. For this series, the convergence interval remains \([-\infty, \infty]\).
By maintaining the radius of convergence, you can be confident that the power series for the derivative also represents a real function for any real \( x \). Always check the behavior at endpoints when applicable, although in this infinite domain, it remains convergent.

Integral of a Series

Integral of a power series is determined by integrating each term separately. For our cosine series, integrating gives: \[ \int f(x) \, dx = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n+1}}{(2n+1)(2n)!} \].
This process integrates each term, slightly increasing the power of \( x \), and dividing by the new power. Notably, the interval of convergence for the integrated series also stays the same, \([-\infty, \infty]\), because the series originated from \( \cos(x) \), which converges everywhere.
While integrating, carefully consider the endpoints, although it's less of a concern here due to cosine's absolute convergence throughout the entire real line.

Cosine Function Series

The original power series given is the well-known Taylor Series for the cosine function:
\[ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \].
This infinite series representation plays a pivotal role in approximating the cosine function with polynomial expressions. Each term in the series adds successive improvements to the approximation of \( \cos(x) \). The cosine series is known for its property of converging to the function's value for all values of \( x \). This property ensures that calculations like differentiation and integration of the series maintain valid approximations over the entire real line.

Taylor Series Expansion

The concept of Taylor Series Expansion is a powerful tool in calculus. It allows a function to be expressed as an infinite sum of terms determined from the function's derivatives at a single point. In particular, for our function \( f(x) \), the Taylor series expanded around zero (Maclaurin series) provides a means to represent the function with great accuracy over an interval.
Taylor series are not just limited to exponents like \( x^n \); they can be expanded for any smooth function like sine, cosine, or exponential functions.
The utility of this lies in simplifying calculations, providing a finite approximation when needed, and deepening the understanding of how functions behave locally. With the cosine series as an example, one can see Taylor's expansion in action, effectively mirroring the behavior of \( \cos(x) \) across all of its domains.