Convergence and Divergence of Series

  1. Series
    A series is an expression formed by adding the terms of a sequence together, written as: It is represented using the summation symbol as .

  2. Partial Sum
    The sum of the first terms of the series is called the partial sum, denoted by:

  3. Sum of the Series
    If the sequence of partial sums converges to a certain value , that is, then the series is said to converge to , and is called the sum of the series: If the sequence of partial sums diverges, the series is said to diverge.

Relationship Between Series and Sequence Limits

  1. If the series converges, then .
  2. If , then the series diverges.
The converse of 1. does not hold, meaning that even if , the series does not necessarily converge.
For example, , but the series diverges.

Properties of Series

For two convergent series and , with sums and respectively:
  3. where is a constant


The properties of series apply only to convergent series.