Series Definition

A series is the operation of adding an infinite number expressions together. A series is denoted using the summation operator and the index variable k.

The partial sum of a series is denoted as and represents the summation of the first terms of the series. Shown below are the first 3 partial sums of a series and as well general form of .

There are two sequences associated with every series that can easily be confused with one another. The first is the sequence of -th terms.

The second sequence is the sequence of partial sums where each partial sum represented the sum of the first terms of the series.

Example: Sum of Counting Numbers

For example, the series which corresponds to the natural numbers is represented by the expression below.

The sequence of -th terms is represented as

The sequence of partial sums is represented as

Geometric Series

The variable represents the ratio between the terms of the series and represents the first term of the series.

Power Series

A power series centered at is a series in the form below. The terms represent the ceffecients of the series.