Question

Suppose that some derivative of \(f\) can be represented by a power series in \(x-x_{0}\) in an interval about \(x_{0} .\) Show that \(f\) and all its derivatives can also.

Short answer

Answer: Yes, all the derivatives of the function, including the function itself, can be represented by a power series in \((x-x_0)\) within the interval of convergence. This can be shown by successively differentiating or integrating the given power series for the derivative to obtain power series representations for higher and lower derivatives.

Step-by-step solution

Start with a given power series for some derivative of f

Suppose that the nth derivative of f, denoted by \(f^{(n)}(x)\), is represented by a power series in \((x-x_0)\) in an interval about \(x_0\). Let's write the power series representation of \(f^{(n)}(x)\): $$f^{(n)}(x) = \sum_{k=0}^{\infty} a_k (x-x_0)^k$$ where \(a_k\) are the coefficients of the power series, and the sum is over all non-negative integers k.

Show that all higher derivatives can be represented by a power series

Now, let's differentiate \(f^{(n)}(x)\) once more to find \(f^{(n+1)}(x)\). We will use the property of differentiation of a power series within its interval of convergence: $$f^{(n+1)}(x) = \frac{d}{dx} \left(\sum_{k=0}^{\infty} a_k (x-x_0)^k\right) = \sum_{k=0}^{\infty} \frac{d}{dx}\left(a_k (x-x_0)^k\right) = \sum_{k=1}^{\infty} ka_k (x-x_0)^{k-1}$$ We see that \(f^{(n+1)}(x)\) is also represented by a power series in \((x-x_0)\). By induction, we can show that all higher derivatives of f can also be represented by a power series in \((x-x_0)\).

Show that all lower derivatives can be represented by a power series

: Let's find \(f^{(n-1)}(x)\) by integrating \(f^{(n)}(x)\). We will use the property of integration of a power series within its interval of convergence: $$f^{(n-1)}(x) = \int f^{(n)}(x) \, dx = \int \left(\sum_{k=0}^{\infty} a_k (x-x_0)^k\right) dx = C + \sum_{k=0}^{\infty} a_k \int (x-x_0)^k \, dx = C + \sum_{k=0}^{\infty} a_k \frac{(x-x_0)^{k+1}}{k+1}$$ where C is the constant of integration. We see that \(f^{(n-1)}(x)\) is also represented by a power series in \((x-x_0)\). By induction, we can show that all lower derivatives of f can also be represented by a power series in \((x-x_0)\).

Conclude that all derivatives of f can be represented by a power series

In the previous steps, we demonstrated that all higher and lower derivatives of f can be represented by a power series in \((x-x_0)\). Combining these results, we can conclude that all the derivatives of f, including f itself, can be represented by a power series in \((x-x_0)\).

Key concepts

Derivative

A derivative is an essential concept in calculus, representing the rate at which a function changes at any given point. When dealing with power series, the derivative provides a tool to express how functions behave in terms of simpler polynomial-like expansions.

• Consider a function represented by a power series around a point \(x_0\):

\[f(x) = \sum_{k=0}^{\infty} a_k (x-x_0)^k\]By applying differentiation, we can find the derivative of each term of the series:

• First Derivative: It's obtained by differentiating each term individually:\[f'(x) = \sum_{k=1}^{\infty} ka_k (x-x_0)^{k-1}\]
• Higher Derivatives: Further derivatives follow a pattern, by applying the derivative operator repeatedly. Each time, the power of \((x-x_0)\) decreases by one while being multiplied by the power index.

This repetitive process shows that each subsequent derivative itself can be represented as a power series, a crucial insight when dealing with infinite series.

Interval of Convergence

The interval of convergence is the range of values for which a power series converges to the function it represents. Understanding this interval is vital because both differentiation and integration of power series are valid only within this range.

• Power Series Definition: Recall the general form of a power series: \[\sum_{k=0}^{\infty} a_k (x-x_0)^k\]
• Radius of Convergence: Through the ratio or root test, the radius \(R\) can be found, indicating how far \(x\) can be from \(x_0\) while maintaining convergence.
• Interval: This is expressed as \((x_0 - R, x_0 + R)\). Within this interval, the series will behave well both in terms of function representation and operations like differentiation or integration.

For each derivative of the function, the interval remains the same, which ensures consistency across calculations.

Induction

Induction is a powerful mathematical principle used to prove statements, especially in a step-by-step logical fashion. It is particularly useful in the context of derivatives expressed in power series.

• Basis Step: Start by proving that the initial statement is true for a starting value, often \(n=0\) or the first relevant case. For instance, showing that \(f^{(n)}(x)\) for the starting \(n\) is representable by a power series.
• Inductive Step: Assume that if the statement holds for an arbitrary \(n\), it must also hold true for \(n+1\). This step involves proving the case for \(n+1\) based on the assumption for \(n\).
• Conclusion: Through this logical process, we demonstrate that the statement is true for all relevant values. In this exercise, it ensures that all derivatives higher or lower than a known derivative can also take a similar power series form.

This logical tool is crucial in establishing truths in cases where direct computation for every individual instance is impractical.