Sequences and Series, An Introduction

Introduction

When a function's domain consists of the non-negative integers, it is a sequence. Such a sequence has a limit l, if for every ε > 0 thereis m, such that x > m implies |f(x)-l| < ε. Such a sequence is "convergent". Sequences lacking limits are "divergent". Sequences can approach ∞, just like functions, but they are still considered divergent.

Note that sequence and convergence are also defined in point set topology and metric spaces. These definitions are mathematically equivalent to the one given above.

Like functions, sequences can be added, scaled, multiplied, and divided, while the same operations are applied to their limits. We won't prove this here, because the proofs are algebraically equivalent to the proofs you have already seen regarding adding and scaling limits and multiplying and dividing limits.

A series differs from a sequence only in semantics, syntactically they are both functions of the non-negative integers. Unlike sequences, the values of a series are added together to determine its limit. To find the limit, or the sum of the series s, build a new sequence t such that t_j is the sum of s_i as i runs from 1 to j. These are called partial sums. Note that t_j = t_j-1+s_j for every j. By definition, s converges iff t converges.

As always, the elements of a series could be points in an arbitrary metric space, provided it is possible to add those points together. Usually a series is drawn from the reals or the complex numbers.

Modifying the first n terms of any series does not affect its convergence, it only changes the value of the sum. Effectively, a constant is added to each partial sum t_i for i>n.

If s is a series, multiply all terms by a constant c. The resulting sequence of partial sums is multiplied by c, hence the limit is multiplied by c.

Given two series f and g, let h_j = f_j+g_j. Compute the three sequences of partial sums and verify that the sequence associated with h is indeed the sum of the sequences associated with f and g. Thus series can be added together and their limits are added.

Combine these two results, and a linear combination of convergent series is convergent, and its limit is the linear combination of the original limits.