The Definition of a Series

Let

a[1], a[2], a[3], ...

be a sequence. We call the sum

a[1] + a[2] + a[3] + ... + a[n] + ...

an infinite series (or just a series) and denote it as

the sum over n from 1 to infinity of a[n].

We define a second sequence, s[n], called the partial sums, by

s[1] = a[1],

s[2] = a[1]+a[2],

s[3] = a[1]+a[2]+a[3],

or, in general,

s[n] = the sum over i from 1 to n of a[i].

We then define convergence as follows:

Definition of Convergence, or the Limit of a Series

Given a series

the sum over n from 1 to infinity of a[n] = a[1] + a[2] + a[3] + ... + a[n] + ...

let s[n] denote its nth partial sum:

s[n] = the sum over i from 1 to n of a[i] = a[1] + a[2] + a[3] + ... + a[n].

If the sequence s[n] has a limit, that is, if there is some s such that for all epsilon > 0 there exists some N > 0 such that |s[n] - s| < epsilon, then the series is called convergent, and we say the series converges. We write

a[1] + a[2] + a[3] + ... + a[n] + ... = s

or

the sum over n from 1 to infinity of a[n] = s.

The number s is called the sum of the series. If the series does not converge, the series is called divergent, and we say the series diverges.

Copyright © 1996 Department of Mathematics, Oregon State University

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