Basic Definitions

Definitions for series depend on a knowledge of sequences and their behavior. Recall that a sequence $a:\ensuremath{\mathbb{N} }\to \ensuremath{\mathbb{R} }$ is just a function with domain $\ensuremath{\mathbb{N} }$. The only interesting questions we can ask about such functions relate to boundedness, monotonicity, and limits as $n\to \infty$. A major theorem relates these issues:

Theorem 1.1   Bounded monotone sequences converge.

The proof of this theorem uses the least upper bound axiom for , and thus is rather deep.

A series is the analog of an improper integral:
Definition 1.1   A series $${\sum_{n=0}^\infty a_n}$$ converges to L if and only if the sequence of partial sums $${S_k=\sum_{n=0}^k a_n}$$ converges to L.

We can make use of the convergence of monotone bounded sequences to get a condition for the convergence of series of non-negative terms:
Corollary 1.2   If each $a_n\geq 0$ then $${\sum_{n=0}^\infty a_n}$$ converges if there is an upper bound on the partial sums.
 

The key point here is that having each $a_n\geq 0$ makes the sequence of partial sums monotone nondecreasing.

For certain important examples we can get explicit expressions for the partial sums.

Our object will be to use series notions to represent functions as an infinite analog to a polynomial, a power series:

 Definition 1.2   A power series is a function of the form

$$\sum_{n=0}^\infty a_n x^n.$$

We evaluate such a function by taking the limit of the resulting series of numbers.Our hope is that power series representations of functions will not be too hard to find, that ignoring all but the first few terms will give us good approximations, and that calculus for power series will prove to be as easy as calculus for polynomials.

1999-11-14