Fundamentals of University Mathematics, 3rd Edition

Chapter 18

Sequences and Series

A sequence is a list of terms x₁, x₂, x₃, …. Consider the sequence with x_n = 1/2ⁿ, i.e.

As n increases, the terms approach 0, so we say that 0 is the limit of the sequence. We can form a new sequence from this with terms s_n equal to the sum of the first n terms,

The limit of this new sequence is 1 and we say that the series

has sum 1.

In the first two sections, we consider real sequences, i.e. sequences whose terms ...