Mathematical Methods for Physics and Engineering - Matematica.NET

4Series and limits4.1 SeriesMany examples exist in the physical sciences of situations where we are presentedwith a sum of terms to evaluate. For example, we may wish to add the contributionsfrom successive slits in a diffraction grating to find the total light intensity at aparticular point behind the grating.A series may have either a finite or infinite number of terms. In either case, thesum of the first N terms of a series (often called a partial sum) is writtenS N = u 1 + u 2 + u 3 + ···+ u N ,where the terms of the series u n , n =1, 2, 3,...,N are numbers, that may ingeneral be complex. If the terms are complex then S N will in general be complexalso, and we can write S N = X N + iY N ,whereX N and Y N are the partial sums ofthe real and imaginary parts of each term separately and are therefore real. If aseries has only N terms then the partial sum S N is of course the sum of the series.Sometimes we may encounter series where each term depends on some variable,x, say. In this case the partial sum of the series will depend on the value assumedby x. For example, consider the infinite seriesS(x) =1+x + x22! + x33! + ··· .This is an example of a power series; these are discussed in more detail insection 4.5. It is in fact the Maclaurin expansion of exp x (see subsection 4.6.3).Therefore S(x) =expx and, of course, varies according to the value of thevariable x. A series might just as easily depend on a complex variable z.A general, random sequence of numbers can be described as a series and a sumof the terms found. However, for cases of practical interest, there will usually be115