In mathematics, the term series is typically used to describe an infinite series. An infinite series is the sum of an infinite sequence. Series can either converge or diverge. When a series converges, it is a single value, since it is the sum of an infinite sequence. When a series diverges, it means that the sum either does not exist or is ±∞. Series are used throughout many different fields of study including mathematics (particularly calculus), physics, computer science, statistics, finance, and more.
In order to work with series, it is important to have an understanding of the notation used. The values of a sequence, the sum of which form a series, are referred to as terms or elements. Terms can be numbers, functions, or essentially anything that can be added. "Series" and "infinite series" are often used interchangeably. The "infinite" in infinite series is meant to emphasize that the series contains an infinite number of terms. Since finite series are not usually considered in the context of calculus, any use of the word "series" on this site will mean infinite series unless otherwise specified. Series are commonly represented in two ways: as a sum of variables followed by an ellipsis (...) or with the use of the summation sign, as shown below:
- , where the subscript denotes which term in the series is being represented.
- . This reads as "the sum from n equals one to infinity of a sub n."
As mentioned above, series can be either convergent or divergent. When a series is divergent, the sum of the series cannot be computed. When it is convergent, the series is said to be summable (specifically the sequence is summable), and a value can be assigned by computing the limit of the partial sums of the sequence, where the partial sum of a sequence may be defined as follows:
|S2||=||a1 + a2|
|S3||=||a1 + a2 + a3|
|Sn||=||a1 + a3 + a3 + ... + an|
Then, the limit of the partial sums can be expressed in series notation as:
When the above limit is equal to some real number S, the limit of the partial sums of the sequence, and therefore the series, converges. Otherwise, the series diverges.
Properties of series
There are a number of properties of series that can be used to manipulate series and thereby simplify or alter them in useful ways.
It is always possible to factor a multiplicative constant out of a series. Given some constant c and sequence an, we can factor the constant out of the series as follows:
Addition and subtraction
Given two convergent sequences, an and bn, their series can be added or subtracted as follows:
It is worth noting that series cannot be multiplied in the same way as they are added. In other words:
Instead, the product of two series can be written as,
Although the index used is often n = 0 or n = 1, it is important to note that the index can start at any n. In some cases, given some series and starting index, it is useful to shift the starting index so as to begin the series at a different value. Consider the following series:
In the above series, the starting index is n = 0. If we instead wanted to begin the series at n = 1, we can shift the index as follows:
The above two series are equivalent. Similarly, we could shift the starting index to n = 2, and the resulting equivalent series would be as follows:
In the above examples, we increased the initial value of the index by 1 each time, which resulted in all the n's in the series decreasing by 1. This will always be true: if the index is increased by some value x, n will decrease by the same value x; on the other hand, if the index is decreased by some value x, n will increase by the same value x.
It is also possible to "remove" terms from the series in a manner that can be (very loosely) related to factoring. Consider the following series:
We can "remove" terms from the above series as follows:
We can do this for any number of terms by simply adjusting the starting index of the series for every term removed. For example, removing the first 3 terms results in the following series:
The above property can be generalized in series notation as follows:
The above properties may not always be used, or may not be used frequently in general. However, they are useful to be aware of because there are certainly cases in which manipulating series is necessary or expedient.