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Series expansion is a mathematical technique that can be used for some mathematical functions that have no direct formula, i.e., no equation has been solved for the function. The technique provides a means of approximating the functions and also a means of finding equivalences between such functions. The mathematics of physical science and engineering heavily depends upon series expansions.
A series is a sum of an infinite, ordered list of numbers or functions, i.e., the sum of a first term, a second term, a third term, and so forth, for an infinite list of terms. The limit of the series can be thought of as the sum of all the terms, even though there are infinitely many. In some cases there are ways to figure out the limit, and in other cases summing some number of terms starting with the first gives a reasonable approximation of the limit. An example of a series (where f(n) = 1/2n):
1/2 + 1/4 + 1/8 + 1/16 + ...
which can be written using Σ, signifying a summation, as:
∞ Σ 1/2n n=1
which means:
1/21 + 1/22 + 1/23 + ... (to 1/2∞)
This series's limit is 1, and an intuitive way to see that is to imagine a circle (e.g., of area "1 square foot") on paper in which you shade half of it, then shade half the rest (i.e., an additional quarter of the circle), then half the rest (an additional eighth), etc., and you can see that as long as you continue acting in this pattern, you're getting closer to shading the entire circle. Each step is like adding another term in the above series. The entire circle's area is a limit because however close to that you wish to reach, it is possible to calculate the number of times you must shade a portion of the circle in such manner: e.g., to shade 99% of the circle requires a minimum of 7 such steps, and 99.9% requires 10, 99.99% requires 14, and so forth. An example of a series that expands a function, in which case, each series term is itself a function of x:
3 5 7
x x x
sin x = x + —— + —— + —— ...
3! 5! 7!
The series is chosen as one that has the function (sin x in this case) as its limit. Calculating the first few terms with x = some number yields an approximation of the sine of that number (termed a series approximation).
Among the arithmetic operations you can carry out with series (including series expansions of functions) is add them, term by term: if you add the terms of the first series above (with a limit of one) term by term to a series with a limit of two, the resulting series has a limit of three. Also, given two functions, f(x) and g(x), each expanded as a series (i.e., "sin x" above), the series expansion of the function h(x) = f(x)+g(x) is the term-by-term sum of the expansions of f(x) and g(x).
Mathematicians have found means to discover series that match otherwise difficult-to-deal-with functions, such as the trigonometric functions, and have a collection of known series to help them start such a discovery. Summing just a number of the early terms offers an approximation of the function, as close as you want, by including enough terms. This is the typical method for calculating sine, cosine, and many other functions, and computer routines for these functions generally use this method. In the past, it was common to use lengthy tables of sine values (and computer programs can use this method as well) but such tables were originally created with arithmetic using series expansions.
Physics benefits from the use of series-calculated trigonometric (and other) functions, but also benefits by using series expansions to assist in analyzing and comparing functions associated with physical processes and observations. A type of series commonly used in physics is a Fourier series, which provides an alternate way of offering the same information as a typical function expression: it is useful for analysis and exploration. Also, some types of sensors' output is the equivalent of the Fourier series of the function of interest, and converting from the Fourier series to a function is part of the processing of the data.