7 Sequences and series
7.1 Concepts
A series is a set of terms that is to be summed. The terms can be numbers, variables,
functions, or more complex quantities. A series can be finite, containing a finite
number of terms,
u
11 + 1 u
21 + 1 u
31 +1-1+ 1 u
nor it can be infinite,
u
11 + 1 u
21 + 1 u
31 + 1 u
41 +1-
where the dots mean that the sum is to be extended indefinitely (ad infinitum). The
terms themselves form a sequence. Sequences are discussed in Section 7.2, finite
series in 7.3, infinite series and tests of convergence in 7.4 and 7.5. In Section 7.6 we
discuss how the MacLaurin and Taylor series can be used to represent certain types of
function as power series (‘infinite polynomials’), and in 7.7 how they are used to
obtain approximate values of functions. Some properties of power series are described
in Section 7.8.
Series occur in all branches of the physical sciences, and the representation of
functions as series is an essential tool for the solution of many physical problems.
Some functions, such as the exponential function and other transcendental functions,
are defined as series, as are some important physical quantities; for example, the
partition function in statistical thermodynamics. We will see in Chapters 12–14 that
the differential equations that are important in the physical sciences often have
solutions that can only be represented as series. Approximate and numerical methods
of solution of problems are often based on series. For example, solutions of the
Schrödinger equation are often represented as series, both in formal theory and in
approximate methods such as the method of ‘linear combination of atomic orbitals’
in molecular-orbital theory (LCAO-MO). An important application of series is in the
analysis of wave forms in terms of Fourier series and Fourier transforms; Fourier
analysis is discussed in Chapter 15.
7.2 Sequences
A sequence is an ordered set of terms
u
1,u
2,u
3,=
with a rule that specifies each term. For example, the numbers
1, 3, 5, 7,=