Derivative of Infinite Series Type Problems

When dealing with infinite series, it is often necessary to understand how to differentiate them term by term. This can be particularly useful in various fields of mathematics and physics, such as in the study of power series, Fourier series, or in solving differential equations.

Basics of Infinite Series

An infinite series is a sum of an infinite number of terms. A simple example is the geometric series:

$$ S = a + ar + ar^2 + ar^3 + \ldots $$

where $a$ is the first term and $r$ is the common ratio. If $|r| < 1$, the series converges to $S = \frac{a}{1 - r}$.

Differentiation of Infinite Series

To differentiate an infinite series, we assume that the series is uniformly convergent within the interval of interest, which allows us to differentiate term by term. The derivative of the series is then the series formed by differentiating each term individually.

Power Series

A power series is an infinite series of the form:

$$ f(x) = \sum_{n=0}^{\infty} c_n (x - a)^n $$

where $c_n$ are coefficients and $a$ is the center of the series.

Differentiation

The derivative of a power series is given by:

$$ f'(x) = \sum_{n=1}^{\infty} n c_n (x - a)^{n-1} $$

This is valid within the radius of convergence of the original series.

Fourier Series

A Fourier series is an infinite series used in the analysis of periodic functions and is given by:

$$ f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)) $$

Differentiation

The derivative of a Fourier series is:

$$ f'(x) = \sum_{n=1}^{\infty} (n b_n \cos(nx) - n a_n \sin(nx)) $$

This assumes that the series converges and the function is piecewise smooth.

Table of Differences and Important Points

Property	Power Series	Fourier Series
Form	$\sum c_n (x - a)^n$	$a_0 + \sum (a_n \cos(nx) + b_n \sin(nx))$
Convergence	Within radius of convergence	For piecewise smooth functions
Differentiation	Term by term, decrease power by 1, multiply by n	Term by term, switch sin and cos, multiply by n
Application	Analytic functions, Taylor series	Periodic functions, signal processing

Formulas

• Derivative of a power series:

$$ \frac{d}{dx} \left( \sum_{n=0}^{\infty} c_n (x - a)^n \right) = \sum_{n=1}^{\infty} n c_n (x - a)^{n-1} $$

• Derivative of a Fourier series:

$$ \frac{d}{dx} \left( a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)) \right) = \sum_{n=1}^{\infty} (n b_n \cos(nx) - n a_n \sin(nx)) $$

Examples

Example 1: Differentiating a Power Series

Consider the power series for $e^x$:

$$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$

Differentiating term by term:

$$ \frac{d}{dx} e^x = \sum_{n=1}^{\infty} \frac{n x^{n-1}}{n!} = \sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1)!} = \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x $$

Example 2: Differentiating a Fourier Series

Consider a Fourier series representation of a sawtooth wave:

$$ f(x) = \frac{2}{\pi} - \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(nx) $$

Differentiating term by term:

$$ f'(x) = - \frac{4}{\pi} \sum_{n=1}^{\infty} (-1)^n \cos(nx) $$

This is the Fourier series of the derivative of the sawtooth wave, which is a square wave.

Conclusion

Differentiating infinite series is a powerful tool in mathematical analysis. By understanding the rules and conditions under which series can be differentiated term by term, one can solve a wide range of problems in mathematics and physics. It is crucial to ensure that the series converges and to apply the appropriate formulas for power series or Fourier series as needed.